Faraday's Law (claimed by Amarsaikhan Jargalsaikhan Fall 2019)
focuses on how a time-varying magnetic field produces a "curly" non-Coulomb electric field, thereby inducing an emf.

==Faraday's Law==

Faraday's Law summarizes the ways voltage can be generated as a result of a time-varying magnetic flux. And it gives a way to connect the magnetic and electric fields in a quantifiable way (will elaborate later). Faraday's law is one of four laws in Maxwell's equations. It tells us that in the presence of a time-varying magnetic field or current (which induces a time-varying magnetic field), there is an emf with a magnitude equal to the change in magnetic flux. It serves as a succinct summary of the ways a voltage (or emf) may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with the magnetic field.

We have previously shown that an electromotive force (emf) can be generated by changing the magnetic flux (see week 10, Motional EMF). The emf generated is directly proportional to the negative rate of change of the magnetic flux. The negative sign is a representative of the conservation of energy or Lenz's Law. Basically the magnetic field created by the induced current (induced by the external magnetic field) needs to oppose the external magnetic field otherwise new energy is created which breaks the law of conservation of energy. In the examples provided in the section "Motional Emf" the magnetic flux is changed by changing the areas of the loops. But as per the Magnetic Flux equations, the flux can also be changed by varying the magnitude and/or direction of the applied magnetic field. So what happens when the magnetic field changes and not the area? This is were Faraday comes in. He discovered through his experiments that the Magnetic flux equation is valid no matter how the flux is changing which allowed him to relate the electric and magnetic fields in a new law. Which is named after him as Faraday's Law but Maxwell also wrote down the differential form of the same law before him.

==Curly Electric Field==

[[File:curly.jpg]]

===Mathematical Model===

'''Faraday's Law'''

emf = <math>{\frac{-d{{Φ}}_{mag}}{dt}}</math>

where emf = <math>\oint\vec{E}_{NC}\bullet d\vec{l}</math> and <math>{{Φ}}_{mag}\equiv\int\vec{B}\bullet\hat{n}dA</math>

In other words: The emf along a round-trip is equal to the rate of change of the magnetic flux on the area encircled by the path.

Direction: With the thumb of your right hand pointing in the direction of the ''-dB/dt'', your fingers curl around in the direction of Enc.

The meaning of the minus sign: If the thumb of your right hand points in the direction of ''-dB/dt'' (that is, the opposite of the direction in which the magnetic field is increasing), your fingers curl around in the direction along which the path integral of electric field is positive. Similarly, the direction of the induced current can be explained using Lenz's Law. Lenz's law states that the induced current from the non-Coulombic electric field is induced in such a way that it produces a magnetic field that opposes the first magnetic field to keep the magnetic flux constant.

'''Formal Version of Faraday's Law'''

<math>\oint\vec{E}_{NC}\bullet d\vec{l} = {\frac{-d}{dt}}\int\vec{B}\bullet\hat{n}dA</math> (sign given by right-hand rule)

====Problem Solving Tips====
To find the direction of the curly electric field, one must find the direction of <math> \frac{-dB}{dt} </math>. Do this using the change in magnetic field as the basis of finding the <math> \frac{-dB}{dt} </math>.

The easiest way to do this is to imagine the a vector for the initial magnetic field, and a vector for the final magnetic field. Then, draw the change in magnetic field vector, <math> \Delta \mathbf{B} </math>, and then the negative vector of that change in magnetic field gives <math> \frac{-dB}{dt} </math>:

[[File:neg_change_B_dt.jpg]]

Pointing the thumb of your right hand in the direction of <math> \frac{-dB}{dt} </math> allows you to curl your fingers in the direction of <math> \mathbf{E_{NC}} </math>.

In this chapter we have seen that a changing magnetic flux induces an emf:

[[File:tips5.png]]

according to Faraday’s law of induction. For a conductor which forms a closed loop, the
emf sets up an induced current ''I =|ε|/R'' , where ''R'' is the resistance of the loop. To
compute the induced current and its direction, we follow the procedure below:

1. For the closed loop of area on a plane, define an area vector A and let it point in
the direction of your thumb, for the convenience of applying the right-hand rule later.
Compute the magnetic flux through the loop using

[[File:tips4.png]]

Determine the sign of the magnetic flux [[File:tips3.png]]

2. Evaluate the rate of change of magnetic flux [[File:tips2.png]] . Keep in mind that the change
could be caused by

[[File:tips.png]]

Determine the sign of [[File:tips2.png]]

3. The sign of the induced emf is the opposite of that of [[File:tips2.png]]. The direction of the
induced current can be found by using Lenz’s law or right-hand rule (discussed previously).

==Computational Model==
The following simulations demonstrate Faraday's Law in action.

==More on Faraday's Law==

Moving a magnet near a coil is not the only way to induce an emf in the coil. Another way to induce emf in a coil is to bring another coil with a steady current near the first coil, thereby changing the magnetic field (and flux) surrounding the first coil, inducing an emf and a current. Also, rotating a bar magnet (or coil) near a coil produces a time-varying magnetic field in the coil since rotating the magnet changes the magnetic field in the coil. The key to inducing the emf in the second coil is to change the magnetic field around it somehow, either by bringing an object that has its own magnetic field around that coil, or changing the current in that object, changing its magnetic field.

Faraday's law can be used to calculate motional emf as well. A bar on two current-carrying rails connected by a resistor moves along the rails, using a magnetic force to induce a current in the wire. There is a magnetic field going into the page. One way to calculate the motional emf is to use the [http://www.physicsbook.gatech.edu/Motional_Emf magnetic force], but an easier way is to use Faraday's law.

Faraday's law, using the change in magnetic flux, can be used to find the motional emf, where the changing factor in the magnetic flux is the area of the circuit as the bar moves, while the magnetic field is kept constant.

[[File:motionalemf.jpg]]

==Examples==

===Simple===

[[File:solenoid.ring.jpg|center|alt=Diagram for simple example]]

''Adapted from the'' Matter & Interactions ''textbook, variation of P12 (4th ed)''.

The solenoid radius is 4 cm and the ring radius is 20 cm. B = 0.8 T inside the solenoid and approximately 0 outside the solenoid. What is the magnetic flux through the outer ring?

''Solution:''

Because the magnetic field outside the solenoid is 0, there is no flux between the ring and solenoid. So the flux in the ring is due to the area of the solenoid, so we use the area of the solenoid to find the flux through the outer ring rather than the area of the ring itself:

<math> \phi = BAcos(\theta)</math>

<math>= (0.8 T)(\pi)(0.04 m)^2cos(0) </math>

<math>= 4.02 x 10^{-3} T*m^2 </math>

===Middle===

[[File:rectanglecoilsolenoid.jpg|center|alt=Diagram for simple example]]

''Adapted from the'' Matter & Interactions ''textbook, variation of P27 (4th ed)''.

A very long, tightly wound solenoid has a circular cross-section of radius 2 cm (only a portion of the very long solenoid is shown). The magnetic field outside the solenoid is negligible. Throughout the inside of the solenoid the magnetic field ''B'' is uniform, to the left as shown, but varying with time ''t: B'' = (.06+.02<math>t^2</math>)T. Surrounding the circular solenoid is a loop of 7 turns of wire in the shape of a rectangle 6 cm by 12 cm. The total resistance of the 7-turn loop is 0.2 ohms.

(a) At ''t'' = 2 s, what is the direction of the current in the 7-turn loop? Explain briefly.

(b) At ''t'' = 2 s, what is the magnitude of the current in the 7-turn loop? Explain briefly.

''Solution''

'''(a)''' The direction of the current in the loop is clockwise.

'''(b)'''

B(t) = (.06+.02<math>t^2</math>)

A = (π)(0.02 m)^2 = .00126 <math>m^2</math>

<math>|{ε}| = AN\frac{dB(t)}{dt}</math>

<math>|{ε}|</math> = (.00126 <math>m^2</math>)(7)<math>\frac{d(.06+.02t^2)}{dt}</math> = (.00882)(.02)(2t) = .0003528t

'''At ''t'' = 2 s:'''

<math>|{ε}|</math> = .0003528(2) = .0007056 V

<math>i = \frac{{ε}}{R}</math>

<math>i = \frac{{.0007056 V}}{0.2 ohms}</math> = '''.00353 A'''

===Difficult===

[[File:difficultfaraday.png]]

A square loop (dimensions L⇥L, total resistance R) is located halfway inside a region with uniform magnetic field B0. The magnitude of the magnetic field suddenly begins to increase linearly in time, eventually quadrupling in a time T.

'''(a) What current (magnitude and direction), if any, is induced in the loop at time T?
'''

<math> |emf| = \frac{-{Φ}_{B}}{Δt} = \frac{A(B_f - B_i)}{T} = \frac{L^2(4B_o - B_o)}{T} = \frac{3B_oL^2}{T}</math>

emf = IR = <math>\frac{3B_oL^2}{TR}</math>

'''(b) What net force (magnitude and direction), if any, is induced on the loop at time T?
'''

<math> F_{top} </math> and <math> F_{bottom} </math> cancel out.
<math> F_{left} </math> = 0 because the left side is out of <math> \vec{B} </math> region.

<math> \vec{F}</math> = <math> \vec{F}_{right} </math> = I <math> \vec{L} \times \vec{B} = (ILB)[(\hat{y} \times - \hat{z} )] = \frac{3B_oL^2}{TR}(4B_o L)(- \hat{x}) = \frac{3{B_o}^2 L^3}{TR}(- \hat{x})</math>

'''(c) What net torque (magnitude and direction), if any, is induced on the loop at time T?
'''

<math> \vec{τ} = \vec{μ} \times \vec{B} = 0 </math> because <math>\vec{μ}</math> and <math>\vec{B}</math> are anti-parallel.

==Connectedness==

Faraday's Law is one of Maxwell's equations which describe the essence of electric and magnetic fields. Maxwell's equations effectively summarize and connect all that we have learned throughout the course of Physics 2.

As an electrical engineer, Faraday's Law is relevant to my major.

== Faraday’s Law Applications ==

Physics 2 content has a lot of important concepts that we as engineers can use to make our jobs easier. Whether it be a direct application of a rule or some derivation of a rule. I know I personally struggle with a concept until I get a concrete real life application that I can see the material applied in. This section of the page will discuss how Faraday’s law is applied to concepts that you as students maybe more familiar with your day to day life.

== Hydroelectric Generators ==
Generators create energy by transforming mechanical motion into electrical energy, but hydroelectric generators use the power of falling water to turn a large turbine which is connected to a large magnet. Around this magnet is a large coil of tightly wound wire. The conceptual creation of electricity is the same as Faraday’s Law except alternating current is being produced, but the idea that a changing magnetic field in a coil of wire induces an electromotive force is still the same. The difference is the magnetic field changes sign and flips resulting in the same thing to occur in the induced EMF. Although the calculations here are slightly more difficult the concepts are the same.

== Transformers ==

Transformers use a similar concept for Faraday’s Law but it’s slightly different. The job of a transformer is to either step up or step down the voltage on the power line. Transformers have a constant magnetic field associated with it due to an iron core. The power supply voltage is adjusted by altering the number of turns of wire around the iron core which in turn alters the EMF of the electricity.

Cartoon of Hydroelectric Plant
https://etrical.files.wordpress.com/2009/12/hydrohow.jpg
Turbine Picture
http://theprepperpodcast.com/wp-content/uploads/2016/02/108-All-About-Hydro-Power-Generators-1054x500.jpg
Transformer Diagram https://en.wikipedia.org/wiki/Transformer#/media/File:Transformer3d_col3.svg

==History==

In 1831, eletromagnetic induction was discovered by Michael Faraday.

===Faraday's Law Experiment ===

[[File:experiment.png]]

Faraday showed that no current is registered in the galvanometer when bar magnet is
stationary with respect to the loop. However, a current is induced in the loop when a
relative motion exists between the bar magnet and the loop. In particular, the
galvanometer deflects in one direction as the magnet approaches the loop, and the
opposite direction as it moves away.

Faraday’s experiment demonstrates that an electric current is induced in the loop by
changing the magnetic field. The coil behaves as if it were connected to an emf source.
Experimentally it is found that the induced emf depends on the rate of change of
magnetic flux through the coil.

Test it out yourself [https://phet.colorado.edu/en/simulation/faradays-law here]

==See also==
===Further Readings===

''Matter and Interactions, Volume II: Electric and Magnetic Interactions, 4th Edition''

''The Electric Life of Michael Faraday'' (2009) by Alan Hirshfield

''Electromagnetic Induction Phenomena'' (2012) by D. Schieber

===External Links===

https://www.youtube.com/watch?v=KGTZPTnZBFE

https://www.nde-ed.org/EducationResources/HighSchool/Electricity/electroinduction.htm

http://www.famousscientists.org/michael-faraday/

http://www.bbc.co.uk/history/historic_figures/faraday_michael.shtml

==References==

''Matter and Interactions, 4th Edition''

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

https://files.t-square.gatech.edu/access/content/group/gtc-970b-7c13-52a7-9627-cdc3154438c6/Test%20Preparation/Old%20Test/2212_Test4_Key-1.pdf

https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction

Faraday's Law

2019-11-22T06:33:30Z

Ajargals3:

Faraday's Law (claimed by Amarsaikhan Jargalsaikhan Spring 2018)
focuses on how a time-varying magnetic field produces a "curly" non-Coulomb electric field, thereby inducing an emf.

==Faraday's Law==

Faraday's Law summarizes the ways voltage can be generated as a result of a time-varying magnetic flux. And it gives a way to connect the magnetic and electric fields in a quantifiable way (will elaborate later). Faraday's law is one of four laws in Maxwell's equations. It tells us that in the presence of a time-varying magnetic field or current (which induces a time-varying magnetic field), there is an emf with a magnitude equal to the change in magnetic flux. It serves as a succinct summary of the ways a voltage (or emf) may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with the magnetic field.

We have previously shown that an electromotive force (emf) can be generated by changing the magnetic flux (see week 10, Motional EMF). The emf generated is directly proportional to the negative rate of change of the magnetic flux. The negative sign is a representative of the conservation of energy or Lenz's Law. Basically the magnetic field created by the induced current (induced by the external magnetic field) needs to oppose the external magnetic field otherwise new energy is created which breaks the law of conservation of energy. In the examples provided in the section "Motional Emf" the magnetic flux is changed by changing the areas of the loops. But as per the Magnetic Flux equations, the flux can also be changed by varying the magnitude and/or direction of the applied magnetic field. So what happens when the magnetic field changes and not the area? This is were Faraday comes in. He discovered through his experiments that the Magnetic flux equation is valid no matter how the flux is changing which allowed him to relate the electric and magnetic fields in a new law. Which is named after him as Faraday's Law but Maxwell also wrote down the differential form of the same law before him.

==Curly Electric Field==

[[File:curly.jpg]]

===Mathematical Model===

'''Faraday's Law'''

emf = <math>{\frac{-d{{Φ}}_{mag}}{dt}}</math>

where emf = <math>\oint\vec{E}_{NC}\bullet d\vec{l}</math> and <math>{{Φ}}_{mag}\equiv\int\vec{B}\bullet\hat{n}dA</math>

In other words: The emf along a round-trip is equal to the rate of change of the magnetic flux on the area encircled by the path.

Direction: With the thumb of your right hand pointing in the direction of the ''-dB/dt'', your fingers curl around in the direction of Enc.

The meaning of the minus sign: If the thumb of your right hand points in the direction of ''-dB/dt'' (that is, the opposite of the direction in which the magnetic field is increasing), your fingers curl around in the direction along which the path integral of electric field is positive. Similarly, the direction of the induced current can be explained using Lenz's Law. Lenz's law states that the induced current from the non-Coulombic electric field is induced in such a way that it produces a magnetic field that opposes the first magnetic field to keep the magnetic flux constant.

'''Formal Version of Faraday's Law'''

<math>\oint\vec{E}_{NC}\bullet d\vec{l} = {\frac{-d}{dt}}\int\vec{B}\bullet\hat{n}dA</math> (sign given by right-hand rule)

====Problem Solving Tips====
To find the direction of the curly electric field, one must find the direction of <math> \frac{-dB}{dt} </math>. Do this using the change in magnetic field as the basis of finding the <math> \frac{-dB}{dt} </math>.

The easiest way to do this is to imagine the a vector for the initial magnetic field, and a vector for the final magnetic field. Then, draw the change in magnetic field vector, <math> \Delta \mathbf{B} </math>, and then the negative vector of that change in magnetic field gives <math> \frac{-dB}{dt} </math>:

[[File:neg_change_B_dt.jpg]]

Pointing the thumb of your right hand in the direction of <math> \frac{-dB}{dt} </math> allows you to curl your fingers in the direction of <math> \mathbf{E_{NC}} </math>.

In this chapter we have seen that a changing magnetic flux induces an emf:

[[File:tips5.png]]

according to Faraday’s law of induction. For a conductor which forms a closed loop, the
emf sets up an induced current ''I =|ε|/R'' , where ''R'' is the resistance of the loop. To
compute the induced current and its direction, we follow the procedure below:

1. For the closed loop of area on a plane, define an area vector A and let it point in
the direction of your thumb, for the convenience of applying the right-hand rule later.
Compute the magnetic flux through the loop using

[[File:tips4.png]]

Determine the sign of the magnetic flux [[File:tips3.png]]

2. Evaluate the rate of change of magnetic flux [[File:tips2.png]] . Keep in mind that the change
could be caused by

[[File:tips.png]]

Determine the sign of [[File:tips2.png]]

3. The sign of the induced emf is the opposite of that of [[File:tips2.png]]. The direction of the
induced current can be found by using Lenz’s law or right-hand rule (discussed previously).

==Computational Model==
The following simulations demonstrate Faraday's Law in action.

==More on Faraday's Law==

Moving a magnet near a coil is not the only way to induce an emf in the coil. Another way to induce emf in a coil is to bring another coil with a steady current near the first coil, thereby changing the magnetic field (and flux) surrounding the first coil, inducing an emf and a current. Also, rotating a bar magnet (or coil) near a coil produces a time-varying magnetic field in the coil since rotating the magnet changes the magnetic field in the coil. The key to inducing the emf in the second coil is to change the magnetic field around it somehow, either by bringing an object that has its own magnetic field around that coil, or changing the current in that object, changing its magnetic field.

Faraday's law can be used to calculate motional emf as well. A bar on two current-carrying rails connected by a resistor moves along the rails, using a magnetic force to induce a current in the wire. There is a magnetic field going into the page. One way to calculate the motional emf is to use the [http://www.physicsbook.gatech.edu/Motional_Emf magnetic force], but an easier way is to use Faraday's law.

Faraday's law, using the change in magnetic flux, can be used to find the motional emf, where the changing factor in the magnetic flux is the area of the circuit as the bar moves, while the magnetic field is kept constant.

[[File:motionalemf.jpg]]

==Examples==

===Simple===

[[File:solenoid.ring.jpg|center|alt=Diagram for simple example]]

''Adapted from the'' Matter & Interactions ''textbook, variation of P12 (4th ed)''.

The solenoid radius is 4 cm and the ring radius is 20 cm. B = 0.8 T inside the solenoid and approximately 0 outside the solenoid. What is the magnetic flux through the outer ring?

''Solution:''

Because the magnetic field outside the solenoid is 0, there is no flux between the ring and solenoid. So the flux in the ring is due to the area of the solenoid, so we use the area of the solenoid to find the flux through the outer ring rather than the area of the ring itself:

<math> \phi = BAcos(\theta)</math>

<math>= (0.8 T)(\pi)(0.04 m)^2cos(0) </math>

<math>= 4.02 x 10^{-3} T*m^2 </math>

===Middle===

[[File:rectanglecoilsolenoid.jpg|center|alt=Diagram for simple example]]

''Adapted from the'' Matter & Interactions ''textbook, variation of P27 (4th ed)''.

A very long, tightly wound solenoid has a circular cross-section of radius 2 cm (only a portion of the very long solenoid is shown). The magnetic field outside the solenoid is negligible. Throughout the inside of the solenoid the magnetic field ''B'' is uniform, to the left as shown, but varying with time ''t: B'' = (.06+.02<math>t^2</math>)T. Surrounding the circular solenoid is a loop of 7 turns of wire in the shape of a rectangle 6 cm by 12 cm. The total resistance of the 7-turn loop is 0.2 ohms.

(a) At ''t'' = 2 s, what is the direction of the current in the 7-turn loop? Explain briefly.

(b) At ''t'' = 2 s, what is the magnitude of the current in the 7-turn loop? Explain briefly.

''Solution''

'''(a)''' The direction of the current in the loop is clockwise.

'''(b)'''

B(t) = (.06+.02<math>t^2</math>)

A = (π)(0.02 m)^2 = .00126 <math>m^2</math>

<math>|{ε}| = AN\frac{dB(t)}{dt}</math>

<math>|{ε}|</math> = (.00126 <math>m^2</math>)(7)<math>\frac{d(.06+.02t^2)}{dt}</math> = (.00882)(.02)(2t) = .0003528t

'''At ''t'' = 2 s:'''

<math>|{ε}|</math> = .0003528(2) = .0007056 V

<math>i = \frac{{ε}}{R}</math>

<math>i = \frac{{.0007056 V}}{0.2 ohms}</math> = '''.00353 A'''

===Difficult===

[[File:difficultfaraday.png]]

A square loop (dimensions L⇥L, total resistance R) is located halfway inside a region with uniform magnetic field B0. The magnitude of the magnetic field suddenly begins to increase linearly in time, eventually quadrupling in a time T.

'''(a) What current (magnitude and direction), if any, is induced in the loop at time T?
'''

<math> |emf| = \frac{-{Φ}_{B}}{Δt} = \frac{A(B_f - B_i)}{T} = \frac{L^2(4B_o - B_o)}{T} = \frac{3B_oL^2}{T}</math>

emf = IR = <math>\frac{3B_oL^2}{TR}</math>

'''(b) What net force (magnitude and direction), if any, is induced on the loop at time T?
'''

<math> F_{top} </math> and <math> F_{bottom} </math> cancel out.
<math> F_{left} </math> = 0 because the left side is out of <math> \vec{B} </math> region.

<math> \vec{F}</math> = <math> \vec{F}_{right} </math> = I <math> \vec{L} \times \vec{B} = (ILB)[(\hat{y} \times - \hat{z} )] = \frac{3B_oL^2}{TR}(4B_o L)(- \hat{x}) = \frac{3{B_o}^2 L^3}{TR}(- \hat{x})</math>

'''(c) What net torque (magnitude and direction), if any, is induced on the loop at time T?
'''

<math> \vec{τ} = \vec{μ} \times \vec{B} = 0 </math> because <math>\vec{μ}</math> and <math>\vec{B}</math> are anti-parallel.

==Connectedness==

Faraday's Law is one of Maxwell's equations which describe the essence of electric and magnetic fields. Maxwell's equations effectively summarize and connect all that we have learned throughout the course of Physics 2.

As an electrical engineer, Faraday's Law is relevant to my major.

== Faraday’s Law Applications ==

Physics 2 content has a lot of important concepts that we as engineers can use to make our jobs easier. Whether it be a direct application of a rule or some derivation of a rule. I know I personally struggle with a concept until I get a concrete real life application that I can see the material applied in. This section of the page will discuss how Faraday’s law is applied to concepts that you as students maybe more familiar with your day to day life.

== Hydroelectric Generators ==
Generators create energy by transforming mechanical motion into electrical energy, but hydroelectric generators use the power of falling water to turn a large turbine which is connected to a large magnet. Around this magnet is a large coil of tightly wound wire. The conceptual creation of electricity is the same as Faraday’s Law except alternating current is being produced, but the idea that a changing magnetic field in a coil of wire induces an electromotive force is still the same. The difference is the magnetic field changes sign and flips resulting in the same thing to occur in the induced EMF. Although the calculations here are slightly more difficult the concepts are the same.

== Transformers ==

Transformers use a similar concept for Faraday’s Law but it’s slightly different. The job of a transformer is to either step up or step down the voltage on the power line. Transformers have a constant magnetic field associated with it due to an iron core. The power supply voltage is adjusted by altering the number of turns of wire around the iron core which in turn alters the EMF of the electricity.

Cartoon of Hydroelectric Plant
https://etrical.files.wordpress.com/2009/12/hydrohow.jpg
Turbine Picture
http://theprepperpodcast.com/wp-content/uploads/2016/02/108-All-About-Hydro-Power-Generators-1054x500.jpg
Transformer Diagram https://en.wikipedia.org/wiki/Transformer#/media/File:Transformer3d_col3.svg

==History==

In 1831, eletromagnetic induction was discovered by Michael Faraday.

===Faraday's Law Experiment ===

[[File:experiment.png]]

Faraday showed that no current is registered in the galvanometer when bar magnet is
stationary with respect to the loop. However, a current is induced in the loop when a
relative motion exists between the bar magnet and the loop. In particular, the
galvanometer deflects in one direction as the magnet approaches the loop, and the
opposite direction as it moves away.

Faraday’s experiment demonstrates that an electric current is induced in the loop by
changing the magnetic field. The coil behaves as if it were connected to an emf source.
Experimentally it is found that the induced emf depends on the rate of change of
magnetic flux through the coil.

Test it out yourself [https://phet.colorado.edu/en/simulation/faradays-law here]

==See also==
===Further Readings===

''Matter and Interactions, Volume II: Electric and Magnetic Interactions, 4th Edition''

''The Electric Life of Michael Faraday'' (2009) by Alan Hirshfield

''Electromagnetic Induction Phenomena'' (2012) by D. Schieber

===External Links===

https://www.youtube.com/watch?v=KGTZPTnZBFE

https://www.nde-ed.org/EducationResources/HighSchool/Electricity/electroinduction.htm

http://www.famousscientists.org/michael-faraday/

http://www.bbc.co.uk/history/historic_figures/faraday_michael.shtml

==References==

''Matter and Interactions, 4th Edition''

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

https://files.t-square.gatech.edu/access/content/group/gtc-970b-7c13-52a7-9627-cdc3154438c6/Test%20Preparation/Old%20Test/2212_Test4_Key-1.pdf

https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction

Center of Mass

2019-04-21T00:56:27Z

Ajargals3:

Amarsaikhan Jargalsaikhan (Spring 2019)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.jpg]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumcm.jpg]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:kineticenergy.jpg]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple Problem with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum Ptotal = <-242, 214, 89>, find the velocity of the center of mass.

===Solution===
Starting from the equation
Ptotal = Mtotal(Vcm)

We have Ptotal, which is <-242, 214, 89>, and Mtotal, which is 2 + 5 + 9 = 16.
Substituting these values into the center of mass momentum equation, we see that:

<-242, 214, 89> = (2 + 5 + 9)(Vcm)

and Vcm = <-242, 214, 89> / 16

Therefore, Vcm = <-15.125, 13.375, 5.56>.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. Additionally, Euler's first law of motion states exactly what was what mentioned in the "Velocity, Momentum, and Kinetic Energy with Center of Mass" section. Leonhard Euler formulated his laws of motion after Newton did, and stated that the linear momentum of a multiparticle system (Ptotal) is equal to the total mass of the multiparticle system (Mtotal) times the velocity of the center of mass of the multiparticle system (Vcm). Therefore, Ptotal = Mtotal(Vcm). [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]