Current in an LC Circuit

CLAIMED BY: Kelsey Dobson 12/5/2015

An LC circuit contains an inductor and a capacitor, and because of the inductor's sluggishness and resistance to change the current, the charge in this circuit can oscillate back and forth forever.

The Main Idea

A circuit containing an inductor "L" and a capacitor "C" is called an LC circuit. This circuit can oscillate continuously if the resistance is small. Accordingly, the connecting wires in an LC circuit are low-resistance thick copper wires. Here is the basic idea behind the process of an LC circuit:

A capacitor is charged initially, and then a switch is closed. This connects the capacitor to the inductor. At first, because of the nature of an inductor, it is difficult for charge on the capacitor to flow through the inductor. This is because the inductor opposes all attempts to change the current. However, since an inductor cannot completely prevent a current from changing, there is more current little by little. This drains the capacitor of its charge.

At the moment the capacitor runs out of charge, there is a current in the inductor, and because inductors are sluggish by nature, the current can't immediately change to zero. Therefore, the system does not come to equililbrium and instead increases the charge in the capacitor. When the capacitor is fully charged, it starts to discharge back through the inductor, and the process repeats. Since the circuit will never reach equilibrium and current never reaches zero, this oscillating process repeats forever. Oscillations could reach a stopping point if there is some resistance, but it may still go through multiple cycles before equilibrium is reached.

A Mathematical Model

An LC circuit has an energy conservation rule associated with it. The energy conservation loop rule for an LC circuit is: [math]\displaystyle{ \Delta V_{capacitor} + \Delta V_{inductor} = \frac{1}{C}-L\frac{dI}{dt} = 0 }[/math] where [math]\displaystyle{ Q }[/math] is the charge on the upper plate of the capacitor and [math]\displaystyle{ I }[/math] is the conventional current leaving the upper plate and going through the inductor.

[math]\displaystyle{ dQ/dt }[/math] is the amount of charge flowing off the capacitor every second. This is the same as the current. Because the charge is leaving the capacitor, [math]\displaystyle{ I = -\frac{dQ}{dt} }[/math].

Energy conservation can be re-written as [math]\displaystyle{ \frac{1}{C}Q+L\frac{d^{2}Q}{dt^{2}} = 0 }[/math]

In addition, by substituting Q and its second derivative into the equation, a possible solution of the rewritten energy conservation equation is <math> Q = Q_{i}cos(\frac{1}{sqrt{LC}}t)

Examples

Simple

Emf of an entire solenoid:

B=μ0NI/d emf = |d(mag flux)/dt| = d/dt|μ0NI/d(piR^2)| = μ0NI/d(piR^2)dI/dt emf = N(μ0N/d(piR^2)dI/dt)= (μ0N^2/d(piR^2)dI/dt)

Middling

What is the self inductance of a common solenoid?

-The self inductance is the constant in the inductance formula solved above. This means that the self inductance constant for a solenoid is (μ0N/d(piR^2)

Difficult

What is the self-inductance of a solenoid that has 100 loops, a radius of 5 cm, and is 1 meter long.

- the self inductance would be (μ0100/1(pi.01^2)= 3.95e-8 henries

Connectedness

1. This topic is interesting because solenoids are a very common placed tool. It is important to know how they operate in order to use them to the best of their capabilities.
2. This topic is personally connected to me because of my major. I am a mechanical engineer and am currently enrolled in ME2110, the "robot building class". In this class to assist with our designs, we often used solenoids as deployment mechanism. Being able to learn about this topic while working with the objects hands on in a non-physics environment helped me immensely in my design process and physics career.

History

The history of inductance goes back quite a long time ago and is pretty complicated. In the early 19th century there were actually two scientist discovering inductance in parallel with each other, one in America and one in England. These two scientist names are Joseph Henry and Michael Faraday. Because of this there is no one named founder of inductance, but both of them did receive credit. Inductance now finds itself as one of Faraday's Laws, giving Michael Faraday his due credit. While the units for inductance are "Henries" named after Joseph Henry.

See also

Faraday's Law

Further reading

Information for this page was found in Matter and Interactions Volume II. If you would like to know more about this topic or other topics like it, please reference this book.

External links

Images were found at http://www.calctool.org/CALC/phys/electromagnetism/solenoid

References

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