Physics Book - User contributions [en]

Charging and Discharging a Capacitor

2026-04-27T01:24:25Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

A capacitor in a circuit acts like a charge reservoir. While charging, current flows from the battery and piles charge onto the plates; while discharging, the plates push current back through the resistor until they're empty. In both cases, the current at any instant is set by how much voltage the capacitor still has left to gain or lose, which is why both processes are exponential.

===Charging===

[[File:Charging a capacitor wiki.PNG|center|frame|Charging a capacitor through a resistor (here, a light bulb). The field across the gap drops as charge accumulates, and current decays toward zero.]]

Connect an uncharged capacitor in series with a battery (emf <math>\mathcal{E}</math>) and a resistor. Initially there is no charge on the plates, so the full battery voltage drops across the resistor and the current is at its maximum, <math>I_0 = \mathcal{E}/R</math>. As charge builds up on the plates, it creates a back-voltage <math>V_C = Q/C</math> that opposes the battery. The net driving voltage <math>\mathcal{E} - Q/C</math> shrinks, and so does the current. Charging stops when <math>V_C = \mathcal{E}</math> and <math>Q = C\mathcal{E}</math>.

In the bulb-and-capacitor demo, the bulb is brightest at <math>t = 0</math> and dims to dark as <math>Q \to C\mathcal{E}</math>.

'''Plate polarity.''' Conventional current flows out the + terminal of the battery. The plate it reaches first becomes positive (charge accumulates there); the other plate becomes negative. This determines the sign of <math>V_C</math> when applying Kirchhoff's loop rule.

===Discharging===

[[File:Discharging a capacitor wiki.PNG|center|frame|Discharging a charged capacitor through a resistor. The capacitor itself drives the current, which decays as Q drops.]]

Disconnect the battery and connect the charged capacitor across a resistor. Now the capacitor's voltage <math>V_C = Q/C</math> drives the current: <math>I = V_C/R = Q/(RC)</math>. As <math>Q</math> falls, so does the current — exponentially. The bulb starts bright and dims to dark, just like in charging, but for a different reason: now it's the capacitor running out of charge, not the capacitor opposing the battery.

'''Role of <math>R</math>.''' The resistance sets the timescale through <math>\tau = RC</math>. Larger <math>R</math> ⇒ slower charge/discharge. (A bulb with a thinner filament has higher <math>R</math>, so it lights more dimly but for longer.) Smaller <math>R</math> means faster, brighter, and shorter.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

[[File:RC_circuit_schematic_GHood.png|center|frame|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

[[File:RC_charging_discharging_curves_GHood.png|center|frame|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

==A Computational Model==

The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.

'''Live simulation:''' [https://www.glowscript.org/#/user/gabrielhoodjr/folder/MyPrograms/program/Wiki Run the GlowScript RC-circuit model on glowscript.org] (opens in a new tab; click "Run this program" to start the simulation, then use the ''Charge / Discharge'' button and the sliders for ε, ''R'', and ''C'').

[[File:RC_glowscript_screenshot_GHood.png|center|frame|Screenshot of the GlowScript model running with default values (ε = 9 V, R = 1 kΩ, C = 1 mF, τ = 1 s). The top scene shows the circuit and capacitor; the live readouts beneath give t, Q, I, V<sub>C</sub>, and τ; the bottom plot tracks Q(t)/Q<sub>max</sub> and I(t)/(ε/R) over time. Click the link above to interact with it directly.]]

''For comparison, the textbook analytic Q(t) charging curve is shown below — the GlowScript model's blue trace should reproduce this same exponential approach to Q<sub>max</sub> = Cε.''

[[File:Capacitor Charging.svg|center|frame|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

===Simple===

'''Question 1.''' For a parallel-plate capacitor with capacitance <math>C = \varepsilon_0 A / s</math>, predict how each change affects <math>C</math> and the field <math>E</math> between the plates (the plates carry a fixed charge <math>Q</math>).

(a) Doubling the plate radius:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(b) Doubling the plate radius (effect on <math>E</math> with fixed <math>Q</math>):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

(c) Doubling the plate separation <math>s</math>:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(d) Doubling the capacitance (with <math>Q</math> fixed):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

'''Answers:''' (a) D, (b) A, (c) B, (d) B.

'''Why:'''
* (a) Plate area is <math>A = \pi r^2</math>. Doubling <math>r</math> gives <math>A \to 4A</math>, and since <math>C = \varepsilon_0 A/s</math>, we get <math>C \to 4C</math>.
* (b) The field is <math>E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A)</math>. With <math>Q</math> fixed and <math>A \to 4A</math>, <math>E \to E/4</math>.
* (c) <math>C \propto 1/s</math>, so doubling <math>s</math> halves <math>C</math>.
* (d) With <math>Q</math> fixed, doubling <math>C</math> halves <math>V = Q/C</math>, and since <math>E = V/s</math>, the field also halves.

===Middling===

'''Question 2.''' A battery is connected through a switch to a resistor (light bulb) in series with an initially uncharged capacitor.

[[File:Circuits problem 1 wiki.PNG|center|frame|Series circuit: battery, switch, bulb (resistor), capacitor.]]

What is the current at points A, B, and C (a) the instant the switch closes, and (b) after a long time? What is the final charge on the capacitor?

'''Solution.''' At <math>t=0</math> the capacitor is uncharged, so <math>V_C = 0</math> and the loop equation gives <math>I_0 = \varepsilon/R</math>. Because the circuit is a single series loop, the current is the same at A, B, and C: all equal to <math>\varepsilon/R</math>.

As <math>t \to \infty</math>, the capacitor charges until <math>V_C = \varepsilon</math>. With no voltage across <math>R</math>, the current at every point is zero. The final charge is <math>Q_\infty = C\varepsilon</math>.

[[File:Circuits problem 1 wiki answers.PNG|center|frame|Worked answers for Question 2.]]

===Difficult===

'''Question 3.''' The switch in the circuit below has been closed for a long time, then is opened.

[[File:Circuits problem 2 wiki.PNG|center|frame|Circuit with a battery, capacitor, and bulb; analyze before and after opening the switch.]]

(a) Before the switch opens (steady state): What is the current at each point? What is the charge on the capacitor? Is the bulb lit?

(b) Immediately after the switch opens: Is the bulb lit? After a long time? What is the initial current through the bulb, and in what direction?

'''Solution.'''

(a) In steady state the capacitor is fully charged and acts as an open circuit, so no current flows in the branch containing the capacitor. Current still flows through any branch that bypasses the capacitor (for example the bulb in parallel with the source), set by Ohm's law on that loop. The capacitor charge is <math>Q = C V_C</math>, where <math>V_C</math> equals the steady-state voltage across its terminals.

(b) When the switch opens, the battery is disconnected. The charged capacitor now acts as the only EMF source and discharges through the bulb. Immediately after, the bulb is lit with current <math>I_0 = V_C/R</math>, where <math>V_C</math> is the capacitor voltage just before opening. The current direction reverses relative to the charging direction because the capacitor is now driving the loop. The current decays as <math>I(t) = I_0 e^{-t/RC}</math>, so after several time constants the bulb goes dark.

[[File:Circuits problem 2 wiki answers.PNG|center|frame|Worked answers for Question 3.]]

==Connectedness==

RC circuits show up everywhere in real electronics. The same exponential charge/discharge behavior derived above sets the timing in camera flashes (a capacitor stores energy and dumps it into the bulb), defibrillators (a large capacitor delivers a controlled current pulse), and the smoothing capacitors in any DC power supply. The time constant <math>\tau = RC</math> is also the basic building block for low-pass and high-pass filters in audio gear and signal processing — picking <math>R</math> and <math>C</math> is literally how you choose a cutoff frequency.

Capacitors are not chemical batteries: they store energy in an electric field rather than in chemical bonds, so they can charge and discharge much faster but hold far less total energy per unit mass. That trade-off is why supercapacitors are showing up alongside (not replacing) lithium batteries in electric vehicles and regenerative braking systems, and why pulsed-power applications — from particle accelerators to high-energy physics experiments — rely on capacitor banks rather than batteries.

==History==

The capacitor was discovered independently in 1745–46 by Ewald Georg von Kleist in Germany and Pieter van Musschenbroek at the University of Leiden, the latter giving the device its early name: the Leyden jar. Both used a glass jar partially filled with water, with a wire passing through the stopper, and charged it with an electrostatic generator; touching the wire produced a painful shock as the jar discharged.

Benjamin Franklin investigated the Leyden jar in the 1740s and showed that the stored charge resided on the glass dielectric itself rather than on the water, and he coined the term "battery" for an array of such jars wired together. The modern parallel-plate geometry and the formal capacitance relation ''C'' = ''ε''<sub>0</sub>''A''/''s'' followed in the 19th century alongside the development of electromagnetism by Faraday, Maxwell, and others. The unit of capacitance, the farad, is named for Michael Faraday.

== See also ==

* [[RC Circuit]]
* [[Capacitor]]
* [[Resistors and Conductivity]]
* [[Kirchhoff's Loop Rule]]

===Further reading===

* Williams, Henry Smith. ''A History of Science, Volume II, Part VI: The Leyden Jar Discovered''.
* Keithley, Joseph F. (1999). ''The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s''. IEEE Press.

===External links===

* [https://en.wikipedia.org/wiki/Capacitor Wikipedia: Capacitor]
* [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit]
* [https://www.khanacademy.org/science/physics/circuits-topic Khan Academy: Circuits]
* [https://phet.colorado.edu/en/simulations/capacitor-lab-basics PhET: Capacitor Lab Basics simulation]

==References==

# "Capacitor." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Capacitor
# "RC circuit." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/RC_circuit
# Griffiths, David J. ''Introduction to Electrodynamics''. 4th ed. Cambridge University Press, 2017.

[[Category:Electric Circuits]]

File:RC glowscript screenshot GHood.png

2026-04-27T01:21:51Z

Ghood6: Screenshot of GlowScript RC circuit simulation by Gabriel Hood for the Charging and Discharging a Capacitor wiki page.

== Summary ==
Screenshot of GlowScript RC circuit simulation by Gabriel Hood for the Charging and Discharging a Capacitor wiki page.

Charging and Discharging a Capacitor

2026-04-27T01:05:47Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

A capacitor in a circuit acts like a charge reservoir. While charging, current flows from the battery and piles charge onto the plates; while discharging, the plates push current back through the resistor until they're empty. In both cases, the current at any instant is set by how much voltage the capacitor still has left to gain or lose, which is why both processes are exponential.

===Charging===

[[File:Charging a capacitor wiki.PNG|center|frame|Charging a capacitor through a resistor (here, a light bulb). The field across the gap drops as charge accumulates, and current decays toward zero.]]

Connect an uncharged capacitor in series with a battery (emf <math>\mathcal{E}</math>) and a resistor. Initially there is no charge on the plates, so the full battery voltage drops across the resistor and the current is at its maximum, <math>I_0 = \mathcal{E}/R</math>. As charge builds up on the plates, it creates a back-voltage <math>V_C = Q/C</math> that opposes the battery. The net driving voltage <math>\mathcal{E} - Q/C</math> shrinks, and so does the current. Charging stops when <math>V_C = \mathcal{E}</math> and <math>Q = C\mathcal{E}</math>.

In the bulb-and-capacitor demo, the bulb is brightest at <math>t = 0</math> and dims to dark as <math>Q \to C\mathcal{E}</math>.

'''Plate polarity.''' Conventional current flows out the + terminal of the battery. The plate it reaches first becomes positive (charge accumulates there); the other plate becomes negative. This determines the sign of <math>V_C</math> when applying Kirchhoff's loop rule.

===Discharging===

[[File:Discharging a capacitor wiki.PNG|center|frame|Discharging a charged capacitor through a resistor. The capacitor itself drives the current, which decays as Q drops.]]

Disconnect the battery and connect the charged capacitor across a resistor. Now the capacitor's voltage <math>V_C = Q/C</math> drives the current: <math>I = V_C/R = Q/(RC)</math>. As <math>Q</math> falls, so does the current — exponentially. The bulb starts bright and dims to dark, just like in charging, but for a different reason: now it's the capacitor running out of charge, not the capacitor opposing the battery.

'''Role of <math>R</math>.''' The resistance sets the timescale through <math>\tau = RC</math>. Larger <math>R</math> ⇒ slower charge/discharge. (A bulb with a thinner filament has higher <math>R</math>, so it lights more dimly but for longer.) Smaller <math>R</math> means faster, brighter, and shorter.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

[[File:RC_circuit_schematic_GHood.png|center|frame|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

[[File:RC_charging_discharging_curves_GHood.png|center|frame|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

===A Computational Model===

The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.



''Live simulation pending; sample output of charge buildup is shown below.''

[[File:Capacitor Charging.svg|center|frame|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

===Simple===

'''Question 1.''' For a parallel-plate capacitor with capacitance <math>C = \varepsilon_0 A / s</math>, predict how each change affects <math>C</math> and the field <math>E</math> between the plates (the plates carry a fixed charge <math>Q</math>).

(a) Doubling the plate radius:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(b) Doubling the plate radius (effect on <math>E</math> with fixed <math>Q</math>):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

(c) Doubling the plate separation <math>s</math>:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(d) Doubling the capacitance (with <math>Q</math> fixed):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

'''Answers:''' (a) D, (b) A, (c) B, (d) B.

'''Why:'''
* (a) Plate area is <math>A = \pi r^2</math>. Doubling <math>r</math> gives <math>A \to 4A</math>, and since <math>C = \varepsilon_0 A/s</math>, we get <math>C \to 4C</math>.
* (b) The field is <math>E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A)</math>. With <math>Q</math> fixed and <math>A \to 4A</math>, <math>E \to E/4</math>.
* (c) <math>C \propto 1/s</math>, so doubling <math>s</math> halves <math>C</math>.
* (d) With <math>Q</math> fixed, doubling <math>C</math> halves <math>V = Q/C</math>, and since <math>E = V/s</math>, the field also halves.

===Middling===

'''Question 2.''' A battery is connected through a switch to a resistor (light bulb) in series with an initially uncharged capacitor.

[[File:Circuits problem 1 wiki.PNG|center|frame|Series circuit: battery, switch, bulb (resistor), capacitor.]]

What is the current at points A, B, and C (a) the instant the switch closes, and (b) after a long time? What is the final charge on the capacitor?

'''Solution.''' At <math>t=0</math> the capacitor is uncharged, so <math>V_C = 0</math> and the loop equation gives <math>I_0 = \varepsilon/R</math>. Because the circuit is a single series loop, the current is the same at A, B, and C: all equal to <math>\varepsilon/R</math>.

As <math>t \to \infty</math>, the capacitor charges until <math>V_C = \varepsilon</math>. With no voltage across <math>R</math>, the current at every point is zero. The final charge is <math>Q_\infty = C\varepsilon</math>.

[[File:Circuits problem 1 wiki answers.PNG|center|frame|Worked answers for Question 2.]]

===Difficult===

'''Question 3.''' The switch in the circuit below has been closed for a long time, then is opened.

[[File:Circuits problem 2 wiki.PNG|center|frame|Circuit with a battery, capacitor, and bulb; analyze before and after opening the switch.]]

(a) Before the switch opens (steady state): What is the current at each point? What is the charge on the capacitor? Is the bulb lit?

(b) Immediately after the switch opens: Is the bulb lit? After a long time? What is the initial current through the bulb, and in what direction?

'''Solution.'''

(a) In steady state the capacitor is fully charged and acts as an open circuit, so no current flows in the branch containing the capacitor. Current still flows through any branch that bypasses the capacitor (for example the bulb in parallel with the source), set by Ohm's law on that loop. The capacitor charge is <math>Q = C V_C</math>, where <math>V_C</math> equals the steady-state voltage across its terminals.

(b) When the switch opens, the battery is disconnected. The charged capacitor now acts as the only EMF source and discharges through the bulb. Immediately after, the bulb is lit with current <math>I_0 = V_C/R</math>, where <math>V_C</math> is the capacitor voltage just before opening. The current direction reverses relative to the charging direction because the capacitor is now driving the loop. The current decays as <math>I(t) = I_0 e^{-t/RC}</math>, so after several time constants the bulb goes dark.

[[File:Circuits problem 2 wiki answers.PNG|center|frame|Worked answers for Question 3.]]

==Connectedness==

RC circuits show up everywhere in real electronics. The same exponential charge/discharge behavior derived above sets the timing in camera flashes (a capacitor stores energy and dumps it into the bulb), defibrillators (a large capacitor delivers a controlled current pulse), and the smoothing capacitors in any DC power supply. The time constant <math>\tau = RC</math> is also the basic building block for low-pass and high-pass filters in audio gear and signal processing — picking <math>R</math> and <math>C</math> is literally how you choose a cutoff frequency.

Capacitors are not chemical batteries: they store energy in an electric field rather than in chemical bonds, so they can charge and discharge much faster but hold far less total energy per unit mass. That trade-off is why supercapacitors are showing up alongside (not replacing) lithium batteries in electric vehicles and regenerative braking systems, and why pulsed-power applications — from particle accelerators to high-energy physics experiments — rely on capacitor banks rather than batteries.

==History==

The capacitor was discovered independently in 1745–46 by Ewald Georg von Kleist in Germany and Pieter van Musschenbroek at the University of Leiden, the latter giving the device its early name: the Leyden jar. Both used a glass jar partially filled with water, with a wire passing through the stopper, and charged it with an electrostatic generator; touching the wire produced a painful shock as the jar discharged.

Benjamin Franklin investigated the Leyden jar in the 1740s and showed that the stored charge resided on the glass dielectric itself rather than on the water, and he coined the term "battery" for an array of such jars wired together. The modern parallel-plate geometry and the formal capacitance relation ''C'' = ''ε''<sub>0</sub>''A''/''s'' followed in the 19th century alongside the development of electromagnetism by Faraday, Maxwell, and others. The unit of capacitance, the farad, is named for Michael Faraday.

== See also ==

* [[RC Circuit]]
* [[Capacitor]]
* [[Resistors and Conductivity]]
* [[Kirchhoff's Loop Rule]]

===Further reading===

* Williams, Henry Smith. ''A History of Science, Volume II, Part VI: The Leyden Jar Discovered''.
* Keithley, Joseph F. (1999). ''The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s''. IEEE Press.

===External links===

* [https://en.wikipedia.org/wiki/Capacitor Wikipedia: Capacitor]
* [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit]
* [https://www.khanacademy.org/science/physics/circuits-topic Khan Academy: Circuits]
* [https://phet.colorado.edu/en/simulations/capacitor-lab-basics PhET: Capacitor Lab Basics simulation]

==References==

# "Capacitor." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Capacitor
# "RC circuit." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/RC_circuit
# Griffiths, David J. ''Introduction to Electrodynamics''. 4th ed. Cambridge University Press, 2017.

[[Category:Electric Circuits]]

Charging and Discharging a Capacitor

2026-04-27T01:04:54Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

A capacitor in a circuit acts like a charge reservoir. While charging, current flows from the battery and piles charge onto the plates; while discharging, the plates push current back through the resistor until they're empty. In both cases, the current at any instant is set by how much voltage the capacitor still has left to gain or lose, which is why both processes are exponential.

===Charging===

[[File:Charging a capacitor wiki.PNG|center|frame|Charging a capacitor through a resistor (here, a light bulb). The field across the gap drops as charge accumulates, and current decays toward zero.]]

Connect an uncharged capacitor in series with a battery (emf <math>\mathcal{E}</math>) and a resistor. Initially there is no charge on the plates, so the full battery voltage drops across the resistor and the current is at its maximum, <math>I_0 = \mathcal{E}/R</math>. As charge builds up on the plates, it creates a back-voltage <math>V_C = Q/C</math> that opposes the battery. The net driving voltage <math>\mathcal{E} - Q/C</math> shrinks, and so does the current. Charging stops when <math>V_C = \mathcal{E}</math> and <math>Q = C\mathcal{E}</math>.

In the bulb-and-capacitor demo, the bulb is brightest at <math>t = 0</math> and dims to dark as <math>Q \to C\mathcal{E}</math>.

'''Plate polarity.''' Conventional current flows out the + terminal of the battery. The plate it reaches first becomes positive (charge accumulates there); the other plate becomes negative. This determines the sign of <math>V_C</math> when applying Kirchhoff's loop rule.

===Discharging===

[[File:Discharging a capacitor wiki.PNG|center|frame|Discharging a charged capacitor through a resistor. The capacitor itself drives the current, which decays as Q drops.]]

Disconnect the battery and connect the charged capacitor across a resistor. Now the capacitor's voltage <math>V_C = Q/C</math> drives the current: <math>I = V_C/R = Q/(RC)</math>. As <math>Q</math> falls, so does the current — exponentially. The bulb starts bright and dims to dark, just like in charging, but for a different reason: now it's the capacitor running out of charge, not the capacitor opposing the battery.

'''Role of <math>R</math>.''' The resistance sets the timescale through <math>\tau = RC</math>. Larger <math>R</math> ⇒ slower charge/discharge. (A bulb with a thinner filament has higher <math>R</math>, so it lights more dimly but for longer.) Smaller <math>R</math> means faster, brighter, and shorter.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

[[File:RC_circuit_schematic_GHood.png|center|frame|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

[[File:RC_charging_discharging_curves_GHood.png|center|frame|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

===A Computational Model===

The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.



''Live simulation pending; sample output of charge buildup is shown below.''

[[File:Capacitor Charging.svg|center|frame|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

===Simple===

'''Question 1.''' For a parallel-plate capacitor with capacitance <math>C = \varepsilon_0 A / s</math>, predict how each change affects <math>C</math> and the field <math>E</math> between the plates (the plates carry a fixed charge <math>Q</math>).

(a) Doubling the plate radius:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(b) Doubling the plate radius (effect on <math>E</math> with fixed <math>Q</math>):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

(c) Doubling the plate separation <math>s</math>:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(d) Doubling the capacitance (with <math>Q</math> fixed):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

'''Answers:''' (a) D, (b) A, (c) B, (d) B.

'''Why:'''
* (a) Plate area is <math>A = \pi r^2</math>. Doubling <math>r</math> gives <math>A \to 4A</math>, and since <math>C = \varepsilon_0 A/s</math>, we get <math>C \to 4C</math>.
* (b) The field is <math>E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A)</math>. With <math>Q</math> fixed and <math>A \to 4A</math>, <math>E \to E/4</math>.
* (c) <math>C \propto 1/s</math>, so doubling <math>s</math> halves <math>C</math>.
* (d) With <math>Q</math> fixed, doubling <math>C</math> halves <math>V = Q/C</math>, and since <math>E = V/s</math>, the field also halves.

===Middling===

'''Question 2.''' A battery is connected through a switch to a resistor (light bulb) in series with an initially uncharged capacitor.

[[File:Circuits problem 1 wiki.PNG|center|frame|Series circuit: battery, switch, bulb (resistor), capacitor.]]

What is the current at points A, B, and C (a) the instant the switch closes, and (b) after a long time? What is the final charge on the capacitor?

'''Solution.''' At <math>t=0</math> the capacitor is uncharged, so <math>V_C = 0</math> and the loop equation gives <math>I_0 = \varepsilon/R</math>. Because the circuit is a single series loop, the current is the same at A, B, and C: all equal to <math>\varepsilon/R</math>.

As <math>t \to \infty</math>, the capacitor charges until <math>V_C = \varepsilon</math>. With no voltage across <math>R</math>, the current at every point is zero. The final charge is <math>Q_\infty = C\varepsilon</math>.

[[File:Circuits problem 1 wiki answers.PNG|center|frame|Worked answers for Question 2.]]

===Difficult===

'''Question 3.''' The switch in the circuit below has been closed for a long time, then is opened.

[[File:Circuits problem 2 wiki.PNG|center|frame|Circuit with a battery, capacitor, and bulb; analyze before and after opening the switch.]]

(a) Before the switch opens (steady state): What is the current at each point? What is the charge on the capacitor? Is the bulb lit?

(b) Immediately after the switch opens: Is the bulb lit? After a long time? What is the initial current through the bulb, and in what direction?

'''Solution.'''

(a) In steady state the capacitor is fully charged and acts as an open circuit, so no current flows in the branch containing the capacitor. Current still flows through any branch that bypasses the capacitor (for example the bulb in parallel with the source), set by Ohm's law on that loop. The capacitor charge is <math>Q = C V_C</math>, where <math>V_C</math> equals the steady-state voltage across its terminals.

(b) When the switch opens, the battery is disconnected. The charged capacitor now acts as the only EMF source and discharges through the bulb. Immediately after, the bulb is lit with current <math>I_0 = V_C/R</math>, where <math>V_C</math> is the capacitor voltage just before opening. The current direction reverses relative to the charging direction because the capacitor is now driving the loop. The current decays as <math>I(t) = I_0 e^{-t/RC}</math>, so after several time constants the bulb goes dark.

[[File:Circuits problem 2 wiki answers.PNG|center|frame|Worked answers for Question 3.]]

==Connectedness==

RC circuits show up everywhere in real electronics. The same exponential charge/discharge behavior derived above sets the timing in camera flashes (a capacitor stores energy and dumps it into the bulb), defibrillators (a large capacitor delivers a controlled current pulse), and the smoothing capacitors in any DC power supply. The time constant <math>\tau = RC</math> is also the basic building block for low-pass and high-pass filters in audio gear and signal processing — picking <math>R</math> and <math>C</math> is literally how you choose a cutoff frequency.

Capacitors are not chemical batteries: they store energy in an electric field rather than in chemical bonds, so they can charge and discharge much faster but hold far less total energy per unit mass. That trade-off is why supercapacitors are showing up alongside (not replacing) lithium batteries in electric vehicles and regenerative braking systems, and why pulsed-power applications — from particle accelerators to high-energy physics experiments — rely on capacitor banks rather than batteries.

==History==

The capacitor was discovered independently in 1745–46 by Ewald Georg von Kleist in Germany and Pieter van Musschenbroek at the University of Leiden, the latter giving the device its early name: the Leyden jar. Both used a glass jar partially filled with water, with a wire passing through the stopper, and charged it with an electrostatic generator; touching the wire produced a painful shock as the jar discharged.

Benjamin Franklin investigated the Leyden jar in the 1740s and showed that the stored charge resided on the glass dielectric itself rather than on the water, and he coined the term "battery" for an array of such jars wired together. The modern parallel-plate geometry and the formal capacitance relation <math>C = \varepsilon_0 A / s</math> followed in the 19th century alongside the development of electromagnetism by Faraday, Maxwell, and others. The unit of capacitance, the farad, is named for Michael Faraday.

== See also ==

* [[RC Circuit]]
* [[Capacitor]]
* [[Resistors and Conductivity]]
* [[Kirchhoff's Loop Rule]]

===Further reading===

* Williams, Henry Smith. ''A History of Science, Volume II, Part VI: The Leyden Jar Discovered''.
* Keithley, Joseph F. (1999). ''The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s''. IEEE Press.

===External links===

* [https://en.wikipedia.org/wiki/Capacitor Wikipedia: Capacitor]
* [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit]
* [https://www.khanacademy.org/science/physics/circuits-topic Khan Academy: Circuits]
* [https://phet.colorado.edu/en/simulations/capacitor-lab-basics PhET: Capacitor Lab Basics simulation]

==References==

# "Capacitor." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Capacitor
# "RC circuit." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/RC_circuit
# Griffiths, David J. ''Introduction to Electrodynamics''. 4th ed. Cambridge University Press, 2017.

[[Category:Electric Circuits]]

Charging and Discharging a Capacitor

2026-04-27T01:02:01Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

A capacitor in a circuit acts like a charge reservoir. While charging, current flows from the battery and piles charge onto the plates; while discharging, the plates push current back through the resistor until they're empty. In both cases, the current at any instant is set by how much voltage the capacitor still has left to gain or lose, which is why both processes are exponential.

===Charging===

[[File:Charging a capacitor wiki.PNG|center|frame|Charging a capacitor through a resistor (here, a light bulb). The field across the gap drops as charge accumulates, and current decays toward zero.]]

Connect an uncharged capacitor in series with a battery (emf <math>\mathcal{E}</math>) and a resistor. Initially there is no charge on the plates, so the full battery voltage drops across the resistor and the current is at its maximum, <math>I_0 = \mathcal{E}/R</math>. As charge builds up on the plates, it creates a back-voltage <math>V_C = Q/C</math> that opposes the battery. The net driving voltage <math>\mathcal{E} - Q/C</math> shrinks, and so does the current. Charging stops when <math>V_C = \mathcal{E}</math> and <math>Q = C\mathcal{E}</math>.

In the bulb-and-capacitor demo, the bulb is brightest at <math>t = 0</math> and dims to dark as <math>Q \to C\mathcal{E}</math>.

'''Plate polarity.''' Conventional current flows out the + terminal of the battery. The plate it reaches first becomes positive (charge accumulates there); the other plate becomes negative. This determines the sign of <math>V_C</math> when applying Kirchhoff's loop rule.

===Discharging===

[[File:Discharging a capacitor wiki.PNG|center|frame|Discharging a charged capacitor through a resistor. The capacitor itself drives the current, which decays as Q drops.]]

Disconnect the battery and connect the charged capacitor across a resistor. Now the capacitor's voltage <math>V_C = Q/C</math> drives the current: <math>I = V_C/R = Q/(RC)</math>. As <math>Q</math> falls, so does the current — exponentially. The bulb starts bright and dims to dark, just like in charging, but for a different reason: now it's the capacitor running out of charge, not the capacitor opposing the battery.

'''Role of <math>R</math>.''' The resistance sets the timescale through <math>\tau = RC</math>. Larger <math>R</math> ⇒ slower charge/discharge. (A bulb with a thinner filament has higher <math>R</math>, so it lights more dimly but for longer.) Smaller <math>R</math> means faster, brighter, and shorter.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

[[File:RC_circuit_schematic_GHood.png|center|frame|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

[[File:RC_charging_discharging_curves_GHood.png|center|frame|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

===A Computational Model===

The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.



''Live simulation pending; sample output of charge buildup is shown below.''

[[File:Capacitor Charging.svg|center|frame|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

===Simple===

'''Question 1.''' For a parallel-plate capacitor with capacitance <math>C = \varepsilon_0 A / s</math>, predict how each change affects <math>C</math> and the field <math>E</math> between the plates (the plates carry a fixed charge <math>Q</math>).

(a) Doubling the plate radius:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(b) Doubling the plate radius (effect on <math>E</math> with fixed <math>Q</math>):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

(c) Doubling the plate separation <math>s</math>:
:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

(d) Doubling the capacitance (with <math>Q</math> fixed):
:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

'''Answers:''' (a) D, (b) A, (c) B, (d) B.

'''Why:'''
* (a) Plate area is <math>A = \pi r^2</math>. Doubling <math>r</math> gives <math>A \to 4A</math>, and since <math>C = \varepsilon_0 A/s</math>, we get <math>C \to 4C</math>.
* (b) The field is <math>E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A)</math>. With <math>Q</math> fixed and <math>A \to 4A</math>, <math>E \to E/4</math>.
* (c) <math>C \propto 1/s</math>, so doubling <math>s</math> halves <math>C</math>.
* (d) With <math>Q</math> fixed, doubling <math>C</math> halves <math>V = Q/C</math>, and since <math>E = V/s</math>, the field also halves.

===Middling===

'''Question 2.''' A battery is connected through a switch to a resistor (light bulb) in series with an initially uncharged capacitor.

[[File:Circuits problem 1 wiki.PNG|center|frame|Series circuit: battery, switch, bulb (resistor), capacitor.]]

What is the current at points A, B, and C (a) the instant the switch closes, and (b) after a long time? What is the final charge on the capacitor?

'''Solution.''' At <math>t=0</math> the capacitor is uncharged, so <math>V_C = 0</math> and the loop equation gives <math>I_0 = \varepsilon/R</math>. Because the circuit is a single series loop, the current is the same at A, B, and C: all equal to <math>\varepsilon/R</math>.

As <math>t \to \infty</math>, the capacitor charges until <math>V_C = \varepsilon</math>. With no voltage across <math>R</math>, the current at every point is zero. The final charge is <math>Q_\infty = C\varepsilon</math>.

[[File:Circuits problem 1 wiki answers.PNG|center|frame|Worked answers for Question 2.]]

===Difficult===

'''Question 3.''' The switch in the circuit below has been closed for a long time, then is opened.

[[File:Circuits problem 2 wiki.PNG|center|frame|Circuit with a battery, capacitor, and bulb; analyze before and after opening the switch.]]

(a) Before the switch opens (steady state): What is the current at each point? What is the charge on the capacitor? Is the bulb lit?

(b) Immediately after the switch opens: Is the bulb lit? After a long time? What is the initial current through the bulb, and in what direction?

'''Solution.'''

(a) In steady state the capacitor is fully charged and acts as an open circuit, so no current flows in the branch containing the capacitor. Current still flows through any branch that bypasses the capacitor (for example the bulb in parallel with the source), set by Ohm's law on that loop. The capacitor charge is <math>Q = C V_C</math>, where <math>V_C</math> equals the steady-state voltage across its terminals.

(b) When the switch opens, the battery is disconnected. The charged capacitor now acts as the only EMF source and discharges through the bulb. Immediately after, the bulb is lit with current <math>I_0 = V_C/R</math>, where <math>V_C</math> is the capacitor voltage just before opening. The current direction reverses relative to the charging direction because the capacitor is now driving the loop. The current decays as <math>I(t) = I_0 e^{-t/RC}</math>, so after several time constants the bulb goes dark.

[[File:Circuits problem 2 wiki answers.PNG|center|frame|Worked answers for Question 3.]]

==Connectedness==

Capacitor can be temporary batteries. Capacitors in parallel can continue to supply current to the circuit if the battery runs out. This is interesting because the capacitor gets its charge from being connected to a chemical battery, but the capacitor itself supplies voltage without chemicals.

Capacitors are being researched for applications in electromagnetic armour and electromagnetic weapons. Currently capacitors are used as detonators in nuclear weapons. Capacitors also are largely involved in separations of AC and DC components.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
In 1745 Ewald Georg von Kleist was the first to "discover" capacitors in Germany. He connected a generator to glass jars of water and charged them. When he touched the wire they were connected to he shocked himself (discharged the capacitor). At the same time Pieter van Musschenbroek made a similar capacitor and named it the Leyden Jar. When Benjamin Franklin studied the Leyden Jar he determined, among other things, that the charge was stored on the glass. During his studies Franklin was the first to give the capacitor the name battery. Since then batteries have most often been electro-chemical cells of capacitors made of sheets of conducting and dielectric material.

== See also ==
===Further reading===
#Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered"
#Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s

===External links===

#Wikipedia Page "Capacitor"[https://en.wikipedia.org/wiki/Capacitor]
#Khan Academy[https://www.khanacademy.org/science/physics/circuits-topic]
#WebAssign "Lab 4 - Charge and Discharge of a Capacitor"[http://www.webassign.net/labsgraceperiod/ncsulcpem2/lab_4/manual.html]
#"Charge of Capacitor vs Current" [http://www.physicsbook.gatech.edu/Charging_and_Discharging_a_Capacitor]

==References==

#Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. 3rd ed. Vol. 2. N.p.: John Wiley and Sons, 2002. Print.
#"Capacitor." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.

[[Category:Which Category did you place this in?]]

Charging and Discharging a Capacitor

2026-04-27T00:44:51Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

A capacitor in a circuit acts like a charge reservoir. While charging, current flows from the battery and piles charge onto the plates; while discharging, the plates push current back through the resistor until they're empty. In both cases, the current at any instant is set by how much voltage the capacitor still has left to gain or lose, which is why both processes are exponential.

===Charging===

[[File:Charging a capacitor wiki.PNG|center|frame|Charging a capacitor through a resistor (here, a light bulb). The field across the gap drops as charge accumulates, and current decays toward zero.]]

Connect an uncharged capacitor in series with a battery (emf <math>\mathcal{E}</math>) and a resistor. Initially there is no charge on the plates, so the full battery voltage drops across the resistor and the current is at its maximum, <math>I_0 = \mathcal{E}/R</math>. As charge builds up on the plates, it creates a back-voltage <math>V_C = Q/C</math> that opposes the battery. The net driving voltage <math>\mathcal{E} - Q/C</math> shrinks, and so does the current. Charging stops when <math>V_C = \mathcal{E}</math> and <math>Q = C\mathcal{E}</math>.

In the bulb-and-capacitor demo, the bulb is brightest at <math>t = 0</math> and dims to dark as <math>Q \to C\mathcal{E}</math>.

'''Plate polarity.''' Conventional current flows out the + terminal of the battery. The plate it reaches first becomes positive (charge accumulates there); the other plate becomes negative. This determines the sign of <math>V_C</math> when applying Kirchhoff's loop rule.

===Discharging===

[[File:Discharging a capacitor wiki.PNG|center|frame|Discharging a charged capacitor through a resistor. The capacitor itself drives the current, which decays as Q drops.]]

Disconnect the battery and connect the charged capacitor across a resistor. Now the capacitor's voltage <math>V_C = Q/C</math> drives the current: <math>I = V_C/R = Q/(RC)</math>. As <math>Q</math> falls, so does the current — exponentially. The bulb starts bright and dims to dark, just like in charging, but for a different reason: now it's the capacitor running out of charge, not the capacitor opposing the battery.

'''Role of <math>R</math>.''' The resistance sets the timescale through <math>\tau = RC</math>. Larger <math>R</math> ⇒ slower charge/discharge. (A bulb with a thinner filament has higher <math>R</math>, so it lights more dimly but for longer.) Smaller <math>R</math> means faster, brighter, and shorter.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

[[File:RC_circuit_schematic_GHood.png|center|frame|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

[[File:RC_charging_discharging_curves_GHood.png|center|frame|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

===A Computational Model===

The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.



''Live simulation pending; sample output of charge buildup is shown below.''

[[File:Capacitor Charging.svg|center|frame|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

'''Question 1'''

Doubling the radius of the capacitor

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the radius of the capacitor

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

Doubling the distance between the plates

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the capacitance

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

ANS: D, B, A, C

'''Question 2'''

[[File:Circuits problem 1 wiki.PNG]]

What is the current at points A,B, and C when the capacitor is not yet charged and when the capacitor is fully charged?

When the capacitor is fully charged what is the charge on the plates?

'''Answer:'''

[[File:Circuits problem 1 wiki answers.PNG]]

'''Question 3'''

[[File:Circuits problem 2 wiki.PNG]]

The switch has been closed for a long time.

What is the current at each point?

What is the charge on the capacitor?

Is the light bulb lit?

The switch is opened.

Immediately after the switch is opened is the bulb lit? After a while?

What current is initially running through the bulb?

Which direction is the current moving?

'''Answer:'''

[[File:Circuits problem 2 wiki answers.PNG]]

==Connectedness==

Capacitor can be temporary batteries. Capacitors in parallel can continue to supply current to the circuit if the battery runs out. This is interesting because the capacitor gets its charge from being connected to a chemical battery, but the capacitor itself supplies voltage without chemicals.

Capacitors are being researched for applications in electromagnetic armour and electromagnetic weapons. Currently capacitors are used as detonators in nuclear weapons. Capacitors also are largely involved in separations of AC and DC components.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
In 1745 Ewald Georg von Kleist was the first to "discover" capacitors in Germany. He connected a generator to glass jars of water and charged them. When he touched the wire they were connected to he shocked himself (discharged the capacitor). At the same time Pieter van Musschenbroek made a similar capacitor and named it the Leyden Jar. When Benjamin Franklin studied the Leyden Jar he determined, among other things, that the charge was stored on the glass. During his studies Franklin was the first to give the capacitor the name battery. Since then batteries have most often been electro-chemical cells of capacitors made of sheets of conducting and dielectric material.

== See also ==
===Further reading===
#Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered"
#Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s

===External links===

#Wikipedia Page "Capacitor"[https://en.wikipedia.org/wiki/Capacitor]
#Khan Academy[https://www.khanacademy.org/science/physics/circuits-topic]
#WebAssign "Lab 4 - Charge and Discharge of a Capacitor"[http://www.webassign.net/labsgraceperiod/ncsulcpem2/lab_4/manual.html]
#"Charge of Capacitor vs Current" [http://www.physicsbook.gatech.edu/Charging_and_Discharging_a_Capacitor]

==References==

#Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. 3rd ed. Vol. 2. N.p.: John Wiley and Sons, 2002. Print.
#"Capacitor." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.

[[Category:Which Category did you place this in?]]

Charging and Discharging a Capacitor

2026-04-27T00:41:32Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

A capacitor in a circuit acts like a charge reservoir. While charging, current flows from the battery and piles charge onto the plates; while discharging, the plates push current back through the resistor until they're empty. In both cases, the current at any instant is set by how much voltage the capacitor still has left to gain or lose, which is why both processes are exponential.

===Charging===

[[File:Charging a capacitor wiki.PNG|center|frame|Charging a capacitor through a resistor (here, a light bulb). The field across the gap drops as charge accumulates, and current decays toward zero.]]

Connect an uncharged capacitor in series with a battery (emf <math>\mathcal{E}</math>) and a resistor. Initially there is no charge on the plates, so the full battery voltage drops across the resistor and the current is at its maximum, <math>I_0 = \mathcal{E}/R</math>. As charge builds up on the plates, it creates a back-voltage <math>V_C = Q/C</math> that opposes the battery. The net driving voltage <math>\mathcal{E} - Q/C</math> shrinks, and so does the current. Charging stops when <math>V_C = \mathcal{E}</math> and <math>Q = C\mathcal{E}</math>.

In the bulb-and-capacitor demo, the bulb is brightest at <math>t = 0</math> and dims to dark as <math>Q \to C\mathcal{E}</math>.

'''Plate polarity.''' Conventional current flows out the + terminal of the battery. The plate it reaches first becomes positive (charge accumulates there); the other plate becomes negative. This determines the sign of <math>V_C</math> when applying Kirchhoff's loop rule.

===Discharging===

[[File:Discharging a capacitor wiki.PNG|center|frame|Discharging a charged capacitor through a resistor. The capacitor itself drives the current, which decays as Q drops.]]

Disconnect the battery and connect the charged capacitor across a resistor. Now the capacitor's voltage <math>V_C = Q/C</math> drives the current: <math>I = V_C/R = Q/(RC)</math>. As <math>Q</math> falls, so does the current — exponentially. The bulb starts bright and dims to dark, just like in charging, but for a different reason: now it's the capacitor running out of charge, not the capacitor opposing the battery.

'''Role of <math>R</math>.''' The resistance sets the timescale through <math>\tau = RC</math>. Larger <math>R</math> ⇒ slower charge/discharge. (A bulb with a thinner filament has higher <math>R</math>, so it lights more dimly but for longer.) Smaller <math>R</math> means faster, brighter, and shorter.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

[[File:RC_circuit_schematic_GHood.png|center|frame|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

[[File:RC_charging_discharging_curves_GHood.png|center|frame|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

===A Computational Model===

[[File:Capacitor Charging.svg|Capacitor Charging]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

'''Question 1'''

Doubling the radius of the capacitor

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the radius of the capacitor

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

Doubling the distance between the plates

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the capacitance

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

ANS: D, B, A, C

'''Question 2'''

[[File:Circuits problem 1 wiki.PNG]]

What is the current at points A,B, and C when the capacitor is not yet charged and when the capacitor is fully charged?

When the capacitor is fully charged what is the charge on the plates?

'''Answer:'''

[[File:Circuits problem 1 wiki answers.PNG]]

'''Question 3'''

[[File:Circuits problem 2 wiki.PNG]]

The switch has been closed for a long time.

What is the current at each point?

What is the charge on the capacitor?

Is the light bulb lit?

The switch is opened.

Immediately after the switch is opened is the bulb lit? After a while?

What current is initially running through the bulb?

Which direction is the current moving?

'''Answer:'''

[[File:Circuits problem 2 wiki answers.PNG]]

==Connectedness==

Capacitor can be temporary batteries. Capacitors in parallel can continue to supply current to the circuit if the battery runs out. This is interesting because the capacitor gets its charge from being connected to a chemical battery, but the capacitor itself supplies voltage without chemicals.

Capacitors are being researched for applications in electromagnetic armour and electromagnetic weapons. Currently capacitors are used as detonators in nuclear weapons. Capacitors also are largely involved in separations of AC and DC components.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
In 1745 Ewald Georg von Kleist was the first to "discover" capacitors in Germany. He connected a generator to glass jars of water and charged them. When he touched the wire they were connected to he shocked himself (discharged the capacitor). At the same time Pieter van Musschenbroek made a similar capacitor and named it the Leyden Jar. When Benjamin Franklin studied the Leyden Jar he determined, among other things, that the charge was stored on the glass. During his studies Franklin was the first to give the capacitor the name battery. Since then batteries have most often been electro-chemical cells of capacitors made of sheets of conducting and dielectric material.

== See also ==
===Further reading===
#Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered"
#Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s

===External links===

#Wikipedia Page "Capacitor"[https://en.wikipedia.org/wiki/Capacitor]
#Khan Academy[https://www.khanacademy.org/science/physics/circuits-topic]
#WebAssign "Lab 4 - Charge and Discharge of a Capacitor"[http://www.webassign.net/labsgraceperiod/ncsulcpem2/lab_4/manual.html]
#"Charge of Capacitor vs Current" [http://www.physicsbook.gatech.edu/Charging_and_Discharging_a_Capacitor]

==References==

#Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. 3rd ed. Vol. 2. N.p.: John Wiley and Sons, 2002. Print.
#"Capacitor." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.

[[Category:Which Category did you place this in?]]

Charging and Discharging a Capacitor

2026-04-27T00:36:07Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

'''Charging a Capacitor'''

[[File:Charging a capacitor wiki.PNG]]

Charging a capacitor isn’t much more difficult than discharging and the same principles still apply. The circuit consists of two batteries, a light bulb, and a capacitor. Essentially, the electron current from the batteries will continue to run until the circuit reaches equilibrium (the capacitor is “full”). Just like when discharging, the bulb starts out bright while the electron current is running, but it slowly dims and goes out as the capacitor charges.

The electron current will flow out the negative end of the battery as usual (conventional current will exit the positive end). Positive charges begin to build up on the right plate and negative charges on the left. The electric field slowly decreases until the net electric field is 0. The fringe field is equal and opposite to the electric field caused by everything else.

If you were to draw a box around the capacitor and label it with positive and negative ends it would look like a battery. It also behaves like a battery. The electron current will continue to flow and the electric field will continue to exist until the potential difference across the capacitor is equal to that of the batteries (sum of emf of all batteries in the circuit).

The following link shows the relationship of capacitor plate charge to current:
[https://trinket.io/glowscript/cd909d0091?outputOnly=true Capacitor Charge Vs Current]

'''Discharging a Capacitor'''

[[File:Discharging a capacitor wiki.PNG]]

A circuit with a charged capacitor has an electric fringe field inside the wire. This field creates an electron current. The electron current will move opposite the direction of the electric field. However, so long as the electron current is running, the capacitor is being discharged. The electron current is moving negative charges away from the negatively charged plate and towards the positively charged plate. Once the charges even out or are neutralized the electric field will cease to exist. Therefore the current stops running.

In the example where the charged capacitor is connected to a light bulb you can see the electric field is large in the beginning but decreases over time. The electron current is also greater in the beginning and decreases over time. Because of this the light bulb starts out shining brightly but slowly dims and goes out.

RIVASH DEEPNARAIN (SPRING 2023):

Determining Which side of the Capacitor becomes Positive and Negative

A common thing that confused me was which side of the capacitor acquires a positive charge and which side is negative. You need to know this because when calculating the voltage across a capacitor, you need to know whether your path goes against the electric field or in the same direction as the electric field that is in between the two plates. Since conventional current, denoted with a capital I, is actually the flow of positive charges, the side that the current meets first will therefore become positive. The other side would then become negative. Examples of this can be seen in the images below. This was confusing to me at first but after I realized this, calculating voltage across capacitors became much simpler.

'''Resistors'''

The amount of resistance in the circuit will determine how long it takes a capacitor to charge or discharge. The less resistance (a light bulb with a thicker filament) the faster the capacitor will charge or discharge. The more resistance (a light bulb with a thin filament) the longer it will take the capacitor to charge or discharge. The thicker filament bulb will be brighter, but won't last as long as a thin filament bulb. '''V = IR''', The larger the resistance the smaller the current.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

[[File:RC_circuit_schematic_GHood.png|center|frame|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

[[File:RC_charging_discharging_curves_GHood.png|center|frame|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

===A Computational Model===

[[File:Capacitor Charging.svg|Capacitor Charging]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

'''Question 1'''

Doubling the radius of the capacitor

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the radius of the capacitor

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

Doubling the distance between the plates

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the capacitance

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

ANS: D, B, A, C

'''Question 2'''

[[File:Circuits problem 1 wiki.PNG]]

What is the current at points A,B, and C when the capacitor is not yet charged and when the capacitor is fully charged?

When the capacitor is fully charged what is the charge on the plates?

'''Answer:'''

[[File:Circuits problem 1 wiki answers.PNG]]

'''Question 3'''

[[File:Circuits problem 2 wiki.PNG]]

The switch has been closed for a long time.

What is the current at each point?

What is the charge on the capacitor?

Is the light bulb lit?

The switch is opened.

Immediately after the switch is opened is the bulb lit? After a while?

What current is initially running through the bulb?

Which direction is the current moving?

'''Answer:'''

[[File:Circuits problem 2 wiki answers.PNG]]

==Connectedness==

Capacitor can be temporary batteries. Capacitors in parallel can continue to supply current to the circuit if the battery runs out. This is interesting because the capacitor gets its charge from being connected to a chemical battery, but the capacitor itself supplies voltage without chemicals.

Capacitors are being researched for applications in electromagnetic armour and electromagnetic weapons. Currently capacitors are used as detonators in nuclear weapons. Capacitors also are largely involved in separations of AC and DC components.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
In 1745 Ewald Georg von Kleist was the first to "discover" capacitors in Germany. He connected a generator to glass jars of water and charged them. When he touched the wire they were connected to he shocked himself (discharged the capacitor). At the same time Pieter van Musschenbroek made a similar capacitor and named it the Leyden Jar. When Benjamin Franklin studied the Leyden Jar he determined, among other things, that the charge was stored on the glass. During his studies Franklin was the first to give the capacitor the name battery. Since then batteries have most often been electro-chemical cells of capacitors made of sheets of conducting and dielectric material.

== See also ==
===Further reading===
#Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered"
#Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s

===External links===

#Wikipedia Page "Capacitor"[https://en.wikipedia.org/wiki/Capacitor]
#Khan Academy[https://www.khanacademy.org/science/physics/circuits-topic]
#WebAssign "Lab 4 - Charge and Discharge of a Capacitor"[http://www.webassign.net/labsgraceperiod/ncsulcpem2/lab_4/manual.html]
#"Charge of Capacitor vs Current" [http://www.physicsbook.gatech.edu/Charging_and_Discharging_a_Capacitor]

==References==

#Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. 3rd ed. Vol. 2. N.p.: John Wiley and Sons, 2002. Print.
#"Capacitor." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.

[[Category:Which Category did you place this in?]]

File:RC charging discharging curves GHood.png

2026-04-27T00:33:38Z

Ghood6: RC charging and discharging curves: normalized charge Q(t)/Q_max vs. time t (in units of τ = RC). Shows charging Q = Q_max(1 − e^(−t/RC)) reaching 63.2% at t = τ, and discharging Q = Q_0 e^(−t/RC) falling to 36.8% at t = τ. Original diagram created for the Charging and Discharging a Capacitor page by Gabriel Hood, Spring 2026. No third-party copyright.

== Summary ==
RC charging and discharging curves: normalized charge Q(t)/Q_max vs. time t (in units of τ = RC). Shows charging Q = Q_max(1 − e^(−t/RC)) reaching 63.2% at t = τ, and discharging Q = Q_0 e^(−t/RC) falling to 36.8% at t = τ. Original diagram created for the Charging and Discharging a Capacitor page by Gabriel Hood, Spring 2026. No third-party copyright.

File:RC circuit schematic GHood.png

2026-04-27T00:32:21Z

Ghood6: Series RC circuit schematic (battery emf ε, switch S, resistor R, capacitor C with charges +Q/-Q, current I = dQ/dt). Original diagram created for the Charging and Discharging a Capacitor page by Gabriel Hood, Spring 2026. No third-party copyright.

== Summary ==
Series RC circuit schematic (battery emf ε, switch S, resistor R, capacitor C with charges +Q/-Q, current I = dQ/dt). Original diagram created for the Charging and Discharging a Capacitor page by Gabriel Hood, Spring 2026. No third-party copyright.

Charging and Discharging a Capacitor

2026-04-27T00:16:22Z

Ghood6:

'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

'''Charging a Capacitor'''

[[File:Charging a capacitor wiki.PNG]]

Charging a capacitor isn’t much more difficult than discharging and the same principles still apply. The circuit consists of two batteries, a light bulb, and a capacitor. Essentially, the electron current from the batteries will continue to run until the circuit reaches equilibrium (the capacitor is “full”). Just like when discharging, the bulb starts out bright while the electron current is running, but it slowly dims and goes out as the capacitor charges.

The electron current will flow out the negative end of the battery as usual (conventional current will exit the positive end). Positive charges begin to build up on the right plate and negative charges on the left. The electric field slowly decreases until the net electric field is 0. The fringe field is equal and opposite to the electric field caused by everything else.

If you were to draw a box around the capacitor and label it with positive and negative ends it would look like a battery. It also behaves like a battery. The electron current will continue to flow and the electric field will continue to exist until the potential difference across the capacitor is equal to that of the batteries (sum of emf of all batteries in the circuit).

The following link shows the relationship of capacitor plate charge to current:
[https://trinket.io/glowscript/cd909d0091?outputOnly=true Capacitor Charge Vs Current]

'''Discharging a Capacitor'''

[[File:Discharging a capacitor wiki.PNG]]

A circuit with a charged capacitor has an electric fringe field inside the wire. This field creates an electron current. The electron current will move opposite the direction of the electric field. However, so long as the electron current is running, the capacitor is being discharged. The electron current is moving negative charges away from the negatively charged plate and towards the positively charged plate. Once the charges even out or are neutralized the electric field will cease to exist. Therefore the current stops running.

In the example where the charged capacitor is connected to a light bulb you can see the electric field is large in the beginning but decreases over time. The electron current is also greater in the beginning and decreases over time. Because of this the light bulb starts out shining brightly but slowly dims and goes out.

RIVASH DEEPNARAIN (SPRING 2023):

Determining Which side of the Capacitor becomes Positive and Negative

A common thing that confused me was which side of the capacitor acquires a positive charge and which side is negative. You need to know this because when calculating the voltage across a capacitor, you need to know whether your path goes against the electric field or in the same direction as the electric field that is in between the two plates. Since conventional current, denoted with a capital I, is actually the flow of positive charges, the side that the current meets first will therefore become positive. The other side would then become negative. Examples of this can be seen in the images below. This was confusing to me at first but after I realized this, calculating voltage across capacitors became much simpler.

'''Resistors'''

The amount of resistance in the circuit will determine how long it takes a capacitor to charge or discharge. The less resistance (a light bulb with a thicker filament) the faster the capacitor will charge or discharge. The more resistance (a light bulb with a thin filament) the longer it will take the capacitor to charge or discharge. The thicker filament bulb will be brighter, but won't last as long as a thin filament bulb. '''V = IR''', The larger the resistance the smaller the current.

===A Mathematical Model===

'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:

<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>

The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so

<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>

Doubling the plate area doubles the capacitance; doubling the gap halves it.

'''Charging derivation.''' For a series RC circuit (battery emf <math>\mathcal{E}</math>, resistor <math>R</math>, initially uncharged capacitor <math>C</math>), Kirchhoff's loop rule gives

<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

Substituting <math>I = dQ/dt</math>:

<math>R\,\frac{dQ}{dt} = \mathcal{E} - \frac{Q}{C} = \frac{C\mathcal{E} - Q}{C}</math>

Separate variables and integrate from <math>(0, 0)</math> to <math>(t, Q)</math>:

<math>\int_0^Q \frac{dQ'}{C\mathcal{E} - Q'} = \int_0^t \frac{dt'}{RC}</math>

<math>-\ln\!\left(\frac{C\mathcal{E} - Q}{C\mathcal{E}}\right) = \frac{t}{RC}</math>

Solving for <math>Q</math>:

<math>Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)</math>

Differentiating gives the current:

<math>I(t) = \frac{dQ}{dt} = \frac{\mathcal{E}}{R}\,e^{-t/RC}</math>

'''Discharging derivation.''' Replace the battery with a wire. The loop rule becomes <math>IR + Q/C = 0</math>, i.e. <math>R\,dQ/dt = -Q/C</math>, which integrates to

<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

===A Computational Model===

[[File:Capacitor Charging.svg|Capacitor Charging]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

'''Question 1'''

Doubling the radius of the capacitor

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the radius of the capacitor

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

Doubling the distance between the plates

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the capacitance

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

ANS: D, B, A, C

'''Question 2'''

[[File:Circuits problem 1 wiki.PNG]]

What is the current at points A,B, and C when the capacitor is not yet charged and when the capacitor is fully charged?

When the capacitor is fully charged what is the charge on the plates?

'''Answer:'''

[[File:Circuits problem 1 wiki answers.PNG]]

'''Question 3'''

[[File:Circuits problem 2 wiki.PNG]]

The switch has been closed for a long time.

What is the current at each point?

What is the charge on the capacitor?

Is the light bulb lit?

The switch is opened.

Immediately after the switch is opened is the bulb lit? After a while?

What current is initially running through the bulb?

Which direction is the current moving?

'''Answer:'''

[[File:Circuits problem 2 wiki answers.PNG]]

==Connectedness==

Capacitor can be temporary batteries. Capacitors in parallel can continue to supply current to the circuit if the battery runs out. This is interesting because the capacitor gets its charge from being connected to a chemical battery, but the capacitor itself supplies voltage without chemicals.

Capacitors are being researched for applications in electromagnetic armour and electromagnetic weapons. Currently capacitors are used as detonators in nuclear weapons. Capacitors also are largely involved in separations of AC and DC components.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
In 1745 Ewald Georg von Kleist was the first to "discover" capacitors in Germany. He connected a generator to glass jars of water and charged them. When he touched the wire they were connected to he shocked himself (discharged the capacitor). At the same time Pieter van Musschenbroek made a similar capacitor and named it the Leyden Jar. When Benjamin Franklin studied the Leyden Jar he determined, among other things, that the charge was stored on the glass. During his studies Franklin was the first to give the capacitor the name battery. Since then batteries have most often been electro-chemical cells of capacitors made of sheets of conducting and dielectric material.

== See also ==
===Further reading===
#Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered"
#Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s

===External links===

#Wikipedia Page "Capacitor"[https://en.wikipedia.org/wiki/Capacitor]
#Khan Academy[https://www.khanacademy.org/science/physics/circuits-topic]
#WebAssign "Lab 4 - Charge and Discharge of a Capacitor"[http://www.webassign.net/labsgraceperiod/ncsulcpem2/lab_4/manual.html]
#"Charge of Capacitor vs Current" [http://www.physicsbook.gatech.edu/Charging_and_Discharging_a_Capacitor]

==References==

#Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. 3rd ed. Vol. 2. N.p.: John Wiley and Sons, 2002. Print.
#"Capacitor." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.

[[Category:Which Category did you place this in?]]

Charging and Discharging a Capacitor

2026-04-27T00:07:02Z

Ghood6:

'''Edited by Gabriel Hood Spring 2025 Physics 2212)'''

A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

==The Main Idea==

'''Charging a Capacitor'''

[[File:Charging a capacitor wiki.PNG]]

Charging a capacitor isn’t much more difficult than discharging and the same principles still apply. The circuit consists of two batteries, a light bulb, and a capacitor. Essentially, the electron current from the batteries will continue to run until the circuit reaches equilibrium (the capacitor is “full”). Just like when discharging, the bulb starts out bright while the electron current is running, but it slowly dims and goes out as the capacitor charges.

The electron current will flow out the negative end of the battery as usual (conventional current will exit the positive end). Positive charges begin to build up on the right plate and negative charges on the left. The electric field slowly decreases until the net electric field is 0. The fringe field is equal and opposite to the electric field caused by everything else.

If you were to draw a box around the capacitor and label it with positive and negative ends it would look like a battery. It also behaves like a battery. The electron current will continue to flow and the electric field will continue to exist until the potential difference across the capacitor is equal to that of the batteries (sum of emf of all batteries in the circuit).

The following link shows the relationship of capacitor plate charge to current:
[https://trinket.io/glowscript/cd909d0091?outputOnly=true Capacitor Charge Vs Current]

'''Discharging a Capacitor'''

[[File:Discharging a capacitor wiki.PNG]]

A circuit with a charged capacitor has an electric fringe field inside the wire. This field creates an electron current. The electron current will move opposite the direction of the electric field. However, so long as the electron current is running, the capacitor is being discharged. The electron current is moving negative charges away from the negatively charged plate and towards the positively charged plate. Once the charges even out or are neutralized the electric field will cease to exist. Therefore the current stops running.

In the example where the charged capacitor is connected to a light bulb you can see the electric field is large in the beginning but decreases over time. The electron current is also greater in the beginning and decreases over time. Because of this the light bulb starts out shining brightly but slowly dims and goes out.

RIVASH DEEPNARAIN (SPRING 2023):

Determining Which side of the Capacitor becomes Positive and Negative

A common thing that confused me was which side of the capacitor acquires a positive charge and which side is negative. You need to know this because when calculating the voltage across a capacitor, you need to know whether your path goes against the electric field or in the same direction as the electric field that is in between the two plates. Since conventional current, denoted with a capital I, is actually the flow of positive charges, the side that the current meets first will therefore become positive. The other side would then become negative. Examples of this can be seen in the images below. This was confusing to me at first but after I realized this, calculating voltage across capacitors became much simpler.

'''Resistors'''

The amount of resistance in the circuit will determine how long it takes a capacitor to charge or discharge. The less resistance (a light bulb with a thicker filament) the faster the capacitor will charge or discharge. The more resistance (a light bulb with a thin filament) the longer it will take the capacitor to charge or discharge. The thicker filament bulb will be brighter, but won't last as long as a thin filament bulb. '''V = IR''', The larger the resistance the smaller the current.

===A Mathematical Model===

<math>V = IR</math>

<math>E = (Q/A)/\varepsilon_0</math>

<math>C = Q/V = \varepsilon_0A/s</math>

<math>V = (Q/A)s/ε0</math>

===A Computational Model===

[[File:Capacitor Charging.svg|Capacitor Charging]]

===Current and Charge within the Capacitors===

The following graphs depict how current and charge within charging and discharging capacitors change over time.

[[File:Priyaaryacapacitorgraphs2.png]]

When the capacitor begins to charge or discharge, current runs through the circuit. It follows logic that whether or not the capacitor is charging or discharging, when the plates begin to reach their equilibrium or zero, respectively, the current slows down to eventually become zero as well.

When the plates are charging or discharging, charge is either accumulating on either sides of the plates (against their natural attractions to the opposite charge) or moving towards the plate of opposite charge. While charging, until the electron current stops running at equilibrium, the charge on the plates will continue to increase until the point of equilibrium, at which point it levels off. Conversely, while discharging, the charge on the plates will continue to decrease until a charge of zero is reached.

'''Time Constant'''

The time constant of a circuit, with units of time, is the product of R and C. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. The following are equations that result in a rough measure of how long it takes charge or current to reach equilibrium.

<math>Q = (C\mathcal{E})[1−e^{(−t/RC)}]</math>

<math>I = (\mathcal{E}/R)[e^{(−t/RC)}]</math>

*Note: <math>\mathcal{E}</math> is electromotive force(emf), whose units are Volts(<math>V</math>)

===The Effect of Surface Area===

[[File:Cap1vscap2.png]]

For two different circuits, each with one of the above capacitors, the circuit with the second capacitor (with more surface area) has a current that stays more constant than the first. The larger capacitor also ends up with a greater amount of charge on its plates.

This is because fringe field magnitude is inversely proportional to plate area, as shown in the equation below. In the first, short time interval, roughly equal quantities of charge will accumulate on the capacitor plates. However, due to its greater area, capacitor 2 will have a weaker fringe field. This, in turn, results in a greater net field for that circuit. This greater net field results in more charge for that circuit compared to the other. More charge will be driven from the negative to the positive plate, and the drift speed changes less for capacitor 2 than capacitor 1.

The equation for fringe electric field is the following:

[[File:Fringe_field_eq.png]]

==Examples==

'''Question 1'''

Doubling the radius of the capacitor

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the radius of the capacitor

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

Doubling the distance between the plates

A) quarters the capacitance

B) halves the capacitance

C) doubles the capacitance

D) quadruples the capacitance

Doubling the capacitance

A) quarters the electric field between the plates

B) halves the electric field between the plates

C) doubles the electric field between the plates

D) quadruples the electric field between the plates

ANS: D, B, A, C

'''Question 2'''

[[File:Circuits problem 1 wiki.PNG]]

What is the current at points A,B, and C when the capacitor is not yet charged and when the capacitor is fully charged?

When the capacitor is fully charged what is the charge on the plates?

'''Answer:'''

[[File:Circuits problem 1 wiki answers.PNG]]

'''Question 3'''

[[File:Circuits problem 2 wiki.PNG]]

The switch has been closed for a long time.

What is the current at each point?

What is the charge on the capacitor?

Is the light bulb lit?

The switch is opened.

Immediately after the switch is opened is the bulb lit? After a while?

What current is initially running through the bulb?

Which direction is the current moving?

'''Answer:'''

[[File:Circuits problem 2 wiki answers.PNG]]

==Connectedness==

Capacitor can be temporary batteries. Capacitors in parallel can continue to supply current to the circuit if the battery runs out. This is interesting because the capacitor gets its charge from being connected to a chemical battery, but the capacitor itself supplies voltage without chemicals.

Capacitors are being researched for applications in electromagnetic armour and electromagnetic weapons. Currently capacitors are used as detonators in nuclear weapons. Capacitors also are largely involved in separations of AC and DC components.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
In 1745 Ewald Georg von Kleist was the first to "discover" capacitors in Germany. He connected a generator to glass jars of water and charged them. When he touched the wire they were connected to he shocked himself (discharged the capacitor). At the same time Pieter van Musschenbroek made a similar capacitor and named it the Leyden Jar. When Benjamin Franklin studied the Leyden Jar he determined, among other things, that the charge was stored on the glass. During his studies Franklin was the first to give the capacitor the name battery. Since then batteries have most often been electro-chemical cells of capacitors made of sheets of conducting and dielectric material.

== See also ==
===Further reading===
#Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered"
#Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s

===External links===

#Wikipedia Page "Capacitor"[https://en.wikipedia.org/wiki/Capacitor]
#Khan Academy[https://www.khanacademy.org/science/physics/circuits-topic]
#WebAssign "Lab 4 - Charge and Discharge of a Capacitor"[http://www.webassign.net/labsgraceperiod/ncsulcpem2/lab_4/manual.html]
#"Charge of Capacitor vs Current" [http://www.physicsbook.gatech.edu/Charging_and_Discharging_a_Capacitor]

==References==

#Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. 3rd ed. Vol. 2. N.p.: John Wiley and Sons, 2002. Print.
#"Capacitor." Wikipedia. Wikimedia Foundation, n.d. Web. 05 Dec. 2015.

[[Category:Which Category did you place this in?]]

Temperature & Entropy

2026-04-26T23:49:47Z

Ghood6: Undo revision 48072 by Ghood6 (talk)

Claimed by Josh Brandt (April 19th, Spring 2022)

<blockquote>
<p>The increase of ... entropy is what distinguishes the past from the future, giving a direction to time.</p>
</blockquote>
<p>— Stephen Hawking</p>

Entropy is a fundamental characteristic of a system: highly related to the topic of Energy. It is often described as a quantitative measure of disorder. Its formal definition makes it a measure of how many ways there is to distribute energy into a system. What makes Entropy so important is its connection to the Second Law of Thermodynamics, which states

<blockquote>
<p>The entropy of a system never decreases spontaneously.</p>
</blockquote>

Entropy is highly related to [[Application of Statistics in Physics]], since it is a result of probability. The Second Law of Thermodynamics rests upon the fact that it is always more likely for entropy to increase, since it is more likely for an outcome to occur if there are more ways which it can. On macroscopic scales, the chances of entropy decrease as non-zero, but are so small as to never be observed.

The concept of Entropy is famous for its proof against the existence of Perpetual Motion Machines; disqualifying their ability to maintain order. The concept has a long and ambiguous history, but in recent science it has taken on a formal and integral definition.

==The Main Idea==

===A Mathematical Model===

The fundamental relationship between Temperature <math>T</math>, Energy <math>E</math> and Entropy <math> S \equiv k_B \ln\Omega</math> is <math>\frac{dS}{dE}=\frac{1}{T} </math>.

In order to understand and predict the behavior of solids, we can employ the Einstein Model. This simple model treats interatomic Coulombic force as a spring, justified by the often highly parabolic potential energy curve near the equilibrium in models of interatomic attraction (see Morse Potential, Buckingham Potential, and Lennard-Jones potential). In this way, one can imagine a cubic lattice of spring-interconnected atoms, with an imaginary cube centered around each one slicing each and every spring in half.

A quantum mechanical harmonic oscillator has quantized energy states, with one quanta being a unit of energy <math>q=\hbar \omega_0 </math>. These quanta can be distributed among a solid by going into anyone of the three half-springs attached to each atom. In fact, the number of ways to distribute <math>q</math> quanta of energy between <math>N</math> oscillators is <math>\Omega = \frac{(q+N-1)!}{q!(N-1)!} </math>. From this, it is clear that entropy is intrinsically related to the number of ways to distribute energy in a system.

From here, it is almost intuitive that systems should evolve to increased entropy over time. Following the fundamental postulate of thermodynamics: in the long-term steady state, all microstates have equal probability, we can see that a macrostate which includes more microstates is the most likely endpoint for a sufficiently time-evolved system. Having many microstates implies having many different ways to distribute energy, and thus high entropy. It should make sense that the Second Law of Thermodynamics states that "the entropy of an isolated system tends to increase through time".

The treatment of a large number of particle is the study of Statistical Mechanics, whose applications to Physics are discussed in another page ([[Application of Statistics in Physics]]). It is apparent that Thermodynamics and Statistical Mechanics have become linked as the atomic behavior behind thermodynamic properties was discovered.

===A Computational Model===

Entropy can be understood by watching the diffusion of gas in the following simulation:

https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html

This simulation connects two essential interpretations of Entropy. Firstly, notice that over time, although the gases begin separated, they end completely mixed. One could easily say that the disorder of the system increased over time, meaning the Entropy increased. Another way to look at it is from the perspective of statistics. There are more configurations where the particles can be disordered than ordered: there's only a couple ways all of the particles can be packed in to one corner, but uncountable ways they can be randomly dispersed. So left to its own, it is exceedingly likely to observe the particles reach a steady-state of perfect diffusion: where they are as mixed as possible. There's simply more ways for this to happen: it is the overwhelmingly likely outcome.

==Examples==

===Simple===

Question: Given that entropy is commonly applied to objects at everyday scales, explain why Boltzmann's definition of entropy is appropriate.

Solution: A pencil has a mass <math>\mathcal{O}(1 \text{ gram})</math>. Pencils are made of wood and graphite, so let's say a pencil is 100% Carbon. Carbon's molar mass is <math>12 \frac{\text{gram}}{\text{mole}} </math>. So there's <math>\frac{1}{12}</math> mole of Carbon atoms in a pencil. Thats <math>\frac{6 \times 10^{23}}{12}</math> Carbon atoms. For each atom, there are three oscillating springs under the Einstein Model, so there is <math>\mathcal{O}(10^{23})</math> oscillators, <math>N</math>. Obviously, <math>\Omega</math> will quickly become unmanageably large. Taking <math>\ln{10^{23}}</math> we get only <math>52</math>. To give appropriate units, we need a constant to multiply, which <math> k_B </math> gives. It should make sense why working with entropy on everyday scales is manageable under this definition.

===Middling===
Question: During a phase transition, a material's temperature does not increase: for example, a solid melting into a liquid or a liquid freezing into a solid. Explain how this is consistent with the equations of entropy.
[[File:Phases.jpeg]]

Solution: Entropy satisfies <math>\frac{dS}{dE} = \frac{1}{T} </math>. When a solid melts into a liquid, there is obviously energy flowing into it, and all things held constant the liquid has more entropy than the solid. So as energy is being dumped in during the transition, entropy is increasing, so <math>\frac{dS}{dE} > 0 </math>. But notice that this does not mean the temperature is increasing. <math>T</math> can be a constant while entropy changes with respect to energy. So this is consistent.

===Difficult===

Question: Show that there are <math>\Omega</math> ways to distribute <math>q</math> indistinguishable quanta between <math>N</math> oscillators where <math>\Omega = \frac{(N+q-1)!}{q!(N-1)!} </math>

Solution:
For illustration, let us first pick the <math>N=3</math> and <math>q=2</math> case and imagine the oscillators as buckets and the quanta as dots. The way we can distribute the quanta into the oscillators is shown below schematically

[[File:Quanta_buckes.jpeg]]

Instead of drawing it out, we can try and create a code that describes each configuration (triple bucket + dot set-up). Let's first imagine creating a "basis", something like this:

[[File:Basis.jpeg]]

With this basis, we could say that the above case 4 can be written as AB. Case 5 is BC and case 6 is AC. Keep in mind that since quanta are indistinguishable, AC and CA are equivalent and so are every other reordering. To designate two dots in one bucket, we can introduce a third letter, D. A D means to add another ball into the bucket in the combination with it, so case 1 is AD. All possible cases are AD, BD, CD, AB, BC, AC. Notice that these are all of the combinations of two letters from the set A,B,C,D.

Now let's take the more general case. If we have <math>N</math> buckets, we need at least <math>N</math> basis buckets. But, each bucket can have <math>q-1</math> extra dots inside, in addition to the single basis bucket. So, to represent all combinations, we need <math>N+q-1</math> "letters". In this scheme, we might say that each additional letter after the basis means to add an extra dot to the letter alphabetically the same with respect to its set to keep things unique. So if the basis is A,B,C then D means add an extra to A, and E would mean add an extra to C.

The amount of ways to distribute the dots is now <math>N+q-1 \choose q </math>, since we can pick sets of <math>q</math> letters from the <math>N+q-1</math> possibilities.

<math>{N+q-1 \choose q} = \frac{(N+q-1)!}{q!(N-1)!}=\Omega</math>

===MATLAB Code===

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% You may change one of these 2 variables, keep in mind once q is over 100

% It may be hard to compute

q = 10; % Quanta of energy

N = 3; % Oscillators

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

f = @(q,N)((factorial(q + (N - 1)))/(factorial(q) * factorial(N-1)));

y = [];

for i = 1:q

y1 = f(i,N);

y = [y y1];

end

x = [1:q];

plot(x,y);

xlim([1 q]);

ylim([0, y(end)]);

xlabel('q (Quanta of Energy)');

ylabel('\Omega (Multiplicity)');

title('Multiplicity vs. Energy Quanta');

box on

grid on;

==Connectedness==
#How is this topic connected to something that you are interested in?

I am very interested in astrophysics, and there are very interesting cosmological analyses of entropy. For example, why do planets and stars form if they're more ordered than the interstellar medium they come from? Is the entropy of the universe really increasing? These topics have implications far beyond classical scales.

#How is it connected to your major?

I'm a Physics major.

#Is there an interesting industrial application?

I'm a Physics major. But, I'll still answer: of course! You might say entropy is the opposing force of efficiency. In history, it was discovered in the context of heat loss with work by Carnot. If we could create infinite-machines, the world would have considerably less problems.

Entropy is highly related to the concept of [[Quantized energy levels]]. The idea that energy can only be distributed in quanta between atoms is essential to understanding the idea of how many ways you can distribute energy in a system. If energy was continuous inside of atoms, there would be infinite ways to distribute energy into any system with more than one "oscillator" in the Einstein Model.

==History==

The concepts of Temperature, Entropy and Energy have been linked throughout history. Historical notions of Heat described it as a particle, such as [[Sir Isaac Newton]] even believing it to have mass. In 1803, Lazare Carnot (father of [[Nicolas Leonard Sadi Carnot]]) formalized the idea that energy cannot be perfectly channeled: disorder is an intrinsic property of energy transformation. In the mid 1800s, [[Rudolf Clausius]] mathematically described a "transformation-content" of energy loss during any thermodynamic process. From the Greek word τροπή pronounced as "tropey" meaning "change", the prefix of "en"ergy was added onto the term when in 1865 entropy as we call it today was introduced. Clausius himself said <blockquote>I prefer going to the ancient languages for the names of important scientific quantities, so that they maymean the same thing in all living tongues. I propose, therefore, to call S the entropy of a body, after the Greek word "transformation". I have designedly coined the word entropy to be similar to energy, for these two quantities are so analogous in their physical significance, that an analogy of denominations seems to me helpful.</blockquote> Two decades later, Ludwig Boltzmann established the connection between entropy and the number of states of a system, introducing the equation we use today and the famous Boltzmann Constant: the first major idea introduced to the modern field of statistical thermodynamics. [1]

==Notable Scientists==
*[[Josiah Willard Gibbs]]
*[[James Prescott Joule]]
*[[Nicolas Leonard Sadi Carnot]]
*[[Rudolf Clausius]]
*[[Ludwig Boltzmann]]

== See also ==

All thermodynamics is linked to the concept of Energy. See also Heat, Temperature, Statistical Mechanics, Spontaneity. For Fun, see the [https://en.wikipedia.org/wiki/Heat_death_of_the_universe Heat Death of the Universe]

===Further reading===

*[[Application of Statistics in Physics]]
*[[Quantized energy levels]]
*[[Electronic Energy Levels and Photons]]

===External links===

*[https://en.wikipedia.org/wiki/Energy Energy]
*[https://en.wikipedia.org/wiki/Thermodynamic_equilibrium Thermodynamics]
*[https://en.wikipedia.org/wiki/Conservation_of_energy Conservation of Energy]
*[https://fs.blog/entropy/ Entropy]

==References==

1. “Entropy.” Wikipedia, Wikimedia Foundation, 23 Apr. 2022, https://en.wikipedia.org/wiki/Entropy

[[Category:Statistical Physics]]

Temperature & Entropy

2026-04-26T23:48:31Z

Ghood6:

<p style="font-size:18pt;"><b>Claimed by Ghood6 (Spring 2026)</b></p>

<blockquote>
<p>The increase of ... entropy is what distinguishes the past from the future, giving a direction to time.</p>
</blockquote>
<p>— Stephen Hawking</p>

Temperature and entropy are two of the most important concepts in thermal physics. '''Temperature''' is a measure of the average kinetic energy associated with the microscopic motion of the particles in a system, and it determines the direction of spontaneous energy transfer between two systems in thermal contact. '''Entropy''' is a quantitative measure of the number of microscopic arrangements (microstates) that correspond to a given macroscopic state of a system. The two are deeply linked: temperature is precisely the inverse of how rapidly the entropy of a system grows when energy is added to it.

What makes entropy so important is its connection to the [https://en.wikipedia.org/wiki/Second_law_of_thermodynamics Second Law of Thermodynamics], which states:

<blockquote>
<p>The total entropy of an isolated system never decreases over time.</p>
</blockquote>

Entropy is highly related to [[Application of Statistics in Physics]], since it is fundamentally a statistical quantity. The Second Law follows from the simple fact that, when many particles share energy, configurations with more microstates are vastly more probable than configurations with few. On macroscopic scales, the probability of spontaneous entropy decrease is so small that it is effectively never observed. The concept of entropy is famous for its proof against the existence of [https://en.wikipedia.org/wiki/Perpetual_motion perpetual motion machines] of the second kind.

==The Main Idea==

Temperature, energy, and entropy together describe the thermal behavior of any system in equilibrium. Imagine two solid blocks at different temperatures placed in contact. Energy will flow from the hotter block to the cooler one, not because energy "knows" which way to go, but because the combined system has many more ways to arrange its energy when it is shared between the blocks at a common temperature than when it is concentrated in one of them. Temperature emerges naturally as the quantity that equalizes when this process is complete.

The microscopic picture used most often in introductory physics is the '''Einstein model of a solid'''. Each atom in the solid is treated as three independent quantum harmonic oscillators (one for each spatial direction), each able to hold energy only in discrete quanta of size <math>q = \hbar \omega_0</math>. The macroscopic properties of the solid — its temperature, its heat capacity, and its entropy — are then computed by counting the number of ways the available energy quanta can be distributed among the oscillators.

===A Mathematical Model===

The fundamental thermodynamic definition of temperature is

<math>\frac{1}{T} = \frac{\partial S}{\partial E}</math>

where ''S'' is the entropy of the system and ''E'' is its internal energy. In words, temperature measures how much the entropy of a system increases per unit of energy added to it. A system that gains a lot of entropy from a small amount of added energy is "cold"; a system whose entropy barely changes when energy is added is "hot."

Boltzmann's statistical definition of entropy is

<math>S = k_B \ln \Omega</math>

where <math>\Omega</math> is the number of microstates consistent with the macrostate of the system, and <math>k_B = 1.38 \times 10^{-23} \text{ J/K}</math> is Boltzmann's constant.

For an Einstein solid containing ''N'' independent quantum oscillators sharing ''q'' indistinguishable energy quanta, the multiplicity is

<math>\Omega(N, q) = \frac{(q + N - 1)!}{q!\,(N - 1)!}</math>

For two Einstein solids in thermal contact with ''q''<sub>total</sub> quanta shared between them, the most probable macrostate is the one that maximizes <math>\Omega_1(N_1, q_1)\,\Omega_2(N_2, q_2)</math> subject to <math>q_1 + q_2 = q_{\text{total}}</math>. Setting the derivative with respect to ''q''<sub>1</sub> to zero gives

<math>\frac{\partial S_1}{\partial E_1} = \frac{\partial S_2}{\partial E_2} \quad\Rightarrow\quad T_1 = T_2</math>

which is exactly the condition of thermal equilibrium. This is the microscopic reason temperatures equalize.

The change in entropy for a reversible process is given by the Clausius definition,

<math>dS = \frac{dQ_{\text{rev}}}{T}</math>

and integrating between two states gives the macroscopic entropy change. For an ideal gas undergoing a reversible isothermal expansion from volume ''V''<sub>1</sub> to ''V''<sub>2</sub>, this becomes

<math>\Delta S = n R \ln\!\left(\frac{V_2}{V_1}\right)</math>

===A Computational Model===

A useful way to build intuition for entropy is to simulate two Einstein solids in thermal contact and watch the most probable energy distribution emerge from random exchanges of quanta. The PhET simulation below also illustrates the same principle for a gas:

* [https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html PhET Gas Properties Simulation] — drag particles around, change temperature and volume, and watch how the system relaxes toward the most probable (highest-multiplicity) state.

A custom GlowScript / VPython model of two coupled Einstein solids is described in the "Suggested Computational Model" section below; it should be embedded here once written.

==Examples==

===Simple===

'''Question:''' A small Einstein solid has ''N'' = 3 oscillators and ''q'' = 4 quanta of energy. How many microstates does the system have, and what is its entropy?

'''Solution:'''
Using the multiplicity formula,

<math>\Omega = \frac{(q + N - 1)!}{q!\,(N - 1)!} = \frac{(4 + 3 - 1)!}{4!\,(3 - 1)!} = \frac{6!}{4!\,2!} = \frac{720}{24 \cdot 2} = 15</math>

The entropy is

<math>S = k_B \ln \Omega = (1.38 \times 10^{-23}\,\text{J/K})\,\ln(15) \approx 3.74 \times 10^{-23}\,\text{J/K}</math>

This is tiny because we only have three oscillators. Real solids have <math>\sim 10^{23}</math> oscillators, which is why measurable entropies are on the order of joules per kelvin.

===Middling===

'''Question:''' During a phase transition, a material's temperature does not increase even as energy is added — for example, while ice melts at 0 °C. Explain how this is consistent with the relation <math>1/T = \partial S / \partial E</math>.

'''Solution:'''
During the transition, energy enters the system as latent heat, and the entropy increases because the liquid phase has many more accessible microstates than the solid phase (molecules are no longer locked into a lattice). Both ''S'' and ''E'' increase, but they increase together at a constant ratio set by the transition temperature, so

<math>\frac{\partial S}{\partial E} = \frac{1}{T_{\text{melt}}} = \text{constant}</math>

The temperature stays fixed at the melting point until the entire phase transition is complete. The latent heat for melting ice, for instance, is

<math>L_f = 334 \text{ J/g}, \qquad \Delta S = \frac{L_f}{T} = \frac{334}{273.15} \approx 1.22 \text{ J/(g·K)}</math>

===Difficult===

'''Question:''' Two Einstein solids ''A'' and ''B'' are in thermal contact. Solid ''A'' has <math>N_A = 300</math> oscillators and solid ''B'' has <math>N_B = 200</math> oscillators. They share <math>q_{\text{total}} = 100</math> quanta. Find the most probable value of <math>q_A</math>, and show that at this point the two solids have the same temperature.

'''Solution:'''
The total multiplicity is

<math>\Omega_{\text{total}}(q_A) = \Omega_A(N_A, q_A)\,\Omega_B(N_B, q_{\text{total}} - q_A)</math>

The most probable macrostate is the one that maximizes <math>\ln \Omega_{\text{total}}</math>. Taking the derivative with respect to <math>q_A</math> and using Stirling's approximation,

<math>\frac{\partial \ln \Omega_A}{\partial q_A} = \frac{\partial \ln \Omega_B}{\partial q_B}</math>

For large ''N'' and ''q'' the Einstein solid satisfies <math>\partial \ln \Omega / \partial q \approx \ln\!\big((q + N)/q\big)</math>, so the equilibrium condition is

<math>\frac{q_A + N_A}{q_A} = \frac{q_B + N_B}{q_B}</math>

Solving with <math>q_A + q_B = 100</math>, <math>N_A = 300</math>, <math>N_B = 200</math> gives

<math>q_A = q_{\text{total}} \cdot \frac{N_A}{N_A + N_B} = 100 \cdot \frac{300}{500} = 60, \qquad q_B = 40</math>

The energy per oscillator is the same in both solids (<math>q_A/N_A = q_B/N_B = 0.2</math>), and since temperature in the Einstein model is a monotonic function of energy per oscillator, the two temperatures are equal — exactly as expected for thermal equilibrium.

==Suggested Computational Model==

Below is a description of a computational model that should be implemented in [https://www.glowscript.org/ GlowScript / VPython] and embedded into this page using a Trinket. The intended behavior is to simulate two Einstein solids exchanging energy quanta and to plot the resulting energy distribution alongside the theoretical multiplicity curve.

'''Goal:''' Visually demonstrate that two Einstein solids in thermal contact spontaneously evolve toward the macrostate of maximum total multiplicity (i.e., thermal equilibrium), and that the equilibrium distribution matches the theoretical prediction <math>q_A/N_A = q_B/N_B</math>.

'''Inputs (constants the user can change at the top of the script):'''
* <code>N_A</code> — number of oscillators in solid A (e.g., 300)
* <code>N_B</code> — number of oscillators in solid B (e.g., 200)
* <code>q_total</code> — total energy quanta to distribute (e.g., 100)
* <code>n_steps</code> — number of Monte Carlo exchange steps (e.g., 50000)
* <code>initial_split</code> — fraction of quanta initially placed in solid A (e.g., 1.0, meaning all quanta start in A)

'''Algorithm:'''
# Create two arrays of length <code>N_A</code> and <code>N_B</code>, each entry representing the number of quanta on that oscillator. Distribute <code>q_total * initial_split</code> quanta randomly among the oscillators of solid A and the rest among solid B.
# At each Monte Carlo step:
## Pick a random oscillator in either solid that currently has at least one quantum.
## Pick a second random oscillator anywhere in either solid.
## Move one quantum from the first to the second. (This conserves <code>q_total</code> and respects the assumption that all microstates are equally likely.)
# Every <code>k</code> steps, record <code>q_A</code> (the total quanta currently in solid A) and update a live histogram of visited <code>q_A</code> values.
# In parallel, plot the theoretical curve <math>\Omega_A(N_A, q_A) \Omega_B(N_B, q_{\text{total}} - q_A)</math> normalized to the same area as the histogram.

'''Visualization:'''
* Two side-by-side rectangular blocks rendered with VPython <code>box</code> objects, each colored on a heat-map scale according to its current energy per oscillator (blue = cold, red = hot).
* A live time series at the top showing <code>q_A(t)</code> approaching the equilibrium value <math>q_A^* = q_{\text{total}} N_A / (N_A + N_B)</math>.
* A live histogram at the bottom of visited <code>q_A</code> values, with the theoretical multiplicity curve overlaid as a smooth line.

'''Expected result:''' Even when all quanta start in solid A, within a few thousand steps the system relaxes to a sharply peaked distribution centered on <math>q_A^* = q_{\text{total}} N_A / (N_A + N_B)</math>, and the histogram matches the theoretical multiplicity curve. This is a direct visualization of the Second Law: the system finds the macrostate of maximum multiplicity.

'''Embedding:''' Once the model is written on glowscript.org and converted to a Trinket, embed it on this page using the Trinket iframe snippet, e.g.

<code><iframe src="https://trinket.io/embed/glowscript/XXXXXXXX" width="100%" height="600" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe></code>

==Connectedness==

'''How is this topic connected to something I am interested in?'''
Entropy connects directly to information theory and machine learning. Claude Shannon's information entropy <math>H = -\sum p_i \log p_i</math> is mathematically identical (up to a constant) to Boltzmann's thermodynamic entropy. Modern machine learning models are routinely trained by minimizing cross-entropy, and the same statistical reasoning that explains why heat flows from hot to cold also explains why a well-trained model converges to the most probable explanation of the data.

'''How is it connected to my major?'''
For engineering majors, the second law sets a hard upper limit on the efficiency of every heat engine and every refrigerator: the [https://en.wikipedia.org/wiki/Carnot_cycle Carnot efficiency] <math>\eta = 1 - T_C/T_H</math>. No amount of cleverness in design can beat this bound. For computer scientists, Landauer's principle states that erasing a single bit of information in an environment at temperature ''T'' must dissipate at least <math>k_B T \ln 2</math> of heat — a thermodynamic cost on computation itself.

'''Industrial applications:'''
Entropy and temperature are central to power generation (steam turbines, gas turbines, nuclear reactors), refrigeration and air conditioning, materials processing, chemical engineering (free energy and reaction spontaneity), and the design of low-power electronics where thermal management is the dominant constraint.

==History==

The concepts of temperature, entropy, and energy have been intertwined throughout the history of physics. Early notions of heat described it as a fluid (the "caloric"), and even Sir Isaac Newton speculated that heat might have mass. In 1824, the French engineer '''Sadi Carnot''' published ''Reflections on the Motive Power of Fire'', which established the upper limit on the efficiency of any heat engine — the first quantitative statement of the second law, written before entropy itself had been named.

In the 1850s, '''Rudolf Clausius''' formalized the idea that thermodynamic processes irreversibly degrade energy. In 1865 he coined the word '''entropy''' from the Greek τροπή ("transformation"), explaining:

<blockquote>
<p>I prefer going to the ancient languages for the names of important scientific quantities, so that they may mean the same thing in all living tongues. I propose, therefore, to call ''S'' the entropy of a body, after the Greek word "transformation". I have designedly coined the word entropy to be similar to energy, for these two quantities are so analogous in their physical significance that an analogy of denominations seems to me helpful.</p>
</blockquote>

In the 1870s and 1880s, '''Ludwig Boltzmann''' established the microscopic foundation of entropy by connecting it to the number of accessible microstates, <math>S = k_B \ln \Omega</math> — the equation engraved on his tombstone in Vienna. '''Josiah Willard Gibbs''' then extended the framework into the modern statistical mechanics used today, treating entropy as a property of probability distributions over microstates.

The 20th century added two more layers. In 1948, '''Claude Shannon''' showed that the same mathematical form governs information. In the 1970s, '''Jacob Bekenstein''' and '''Stephen Hawking''' showed that black holes themselves carry entropy proportional to their horizon area, tying thermodynamics to gravitation and quantum mechanics.

==Notable Scientists==

* Sadi Carnot
* Rudolf Clausius
* James Prescott Joule
* Ludwig Boltzmann
* Josiah Willard Gibbs
* Claude Shannon

==See also==

All thermodynamics is linked to the concept of [[Energy]]. See also [[Heat]], [[Temperature]], [[Statistical Mechanics]], [[Spontaneity]]. For an interesting cosmological perspective, see the [https://en.wikipedia.org/wiki/Heat_death_of_the_universe Heat Death of the Universe].

===Further reading===

* [[Application of Statistics in Physics]]
* [[Quantized energy levels]]
* [[Electronic Energy Levels and Photons]]
* Schroeder, D. V. ''An Introduction to Thermal Physics'' (Addison-Wesley, 2000) — chapters 2 and 3 give the Einstein-solid derivation used above.
* Reif, F. ''Fundamentals of Statistical and Thermal Physics'' (McGraw-Hill, 1965).

===External links===

* [https://en.wikipedia.org/wiki/Entropy Wikipedia: Entropy]
* [https://en.wikipedia.org/wiki/Temperature Wikipedia: Temperature]
* [https://en.wikipedia.org/wiki/Second_law_of_thermodynamics Wikipedia: Second Law of Thermodynamics]
* [https://en.wikipedia.org/wiki/Einstein_solid Wikipedia: Einstein Solid]
* [https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html PhET: Gas Properties]
* [https://www.feynmanlectures.caltech.edu/I_44.html Feynman Lectures, Vol. I, Ch. 44 — The Laws of Thermodynamics]

==References==

# "Entropy." ''Wikipedia'', Wikimedia Foundation, https://en.wikipedia.org/wiki/Entropy
# "Temperature." ''Wikipedia'', Wikimedia Foundation, https://en.wikipedia.org/wiki/Temperature
# "Einstein solid." ''Wikipedia'', Wikimedia Foundation, https://en.wikipedia.org/wiki/Einstein_solid
# Schroeder, Daniel V. ''An Introduction to Thermal Physics''. Addison-Wesley, 2000.
# Carnot, Sadi. ''Reflections on the Motive Power of Fire''. 1824. Public-domain English translation: https://en.wikisource.org/wiki/Reflections_on_the_Motive_Power_of_Heat
# Bekenstein, J. D. "Black Holes and Entropy." ''Physical Review D'' 7, 2333 (1973).

[[Category:Statistical Physics]]