# A Question About Induction: Faraday's Law versus Motional EMF

• MS La Moreaux
MS La Moreaux
TL;DR Summary
Faraday's Law versus Motional Emf
Visualize a rectangular bare wire circuit with one side loose so that it can slide along the adjacent sides and thereby change the size of the circuit. There is a static magnetic field orthogonal to the plane of the circuit and linking the circuit. There will be a motional emf in the moving wire, due to its motion through the magnetic field.

Faraday's Law states that the changing magnetic flux linking the circuit results in an emf in the circuit. This would be in addition to the motional emf, since the two principles are independent of each other. The value of the motional emf is equal to the value given by Faraday's Law, and therefore the total emf would be twice the actual value.

What is the resolution of this contradiction?

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hutchphd
MS La Moreaux said:
TL;DR Summary: Faraday's Law versus Motional Emf

Visualize a rectangular bare wire circuit with one side loose so that it can slide along the adjacent sides and thereby change the size of the circuit. There is a static magnetic field orthogonal to the plane of the circuit and linking the circuit. There will be a motional emf in the moving wire, due to its motion through the magnetic field.

Faraday's Law states that the changing magnetic flux linking the circuit results in an emf in the circuit. This would be in addition to the motional emf, since the two principles are independent of each other. The value of the motional emf is equal to the value given by Faraday's Law, and therefore the total emf would be twice the actual value.

What is the resolution of this contradiction?
Since you have stated in your previous threads that you have a BSEE degree from the University of Michigan, please post your mathematical analysis of this setup. It would help if you could also post a diagram of it, but it's simple enough that we can probably figure it out as long as you define all of your terms and variables in your mathematical analysis. Please be sure to use LaTeX when posting math at PF (see the LaTeX Guide below the Edit window). Thanks.

MS La Moreaux said:
There will be a motional emf in the moving wire, due to its motion through the magnetic field.
EDIT/ADD: Also, if there is no initial current in the wire loop and the B-field piercing the loop is static, what would cause a force to move the 4th side wire of the loop?

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MS La Moreaux said:
Faraday's Law states that the changing magnetic flux linking the circuit results in an emf in the circuit.
You assume a fixed magnetic field, but if you change the area of your rectangular short circuit, the number of lines inside the loop will remain constant, while lines you assume would then be included inside the loop, will actually be pushed away as the perfect conductor moves.

hutchphd
emf = - d(phi)/dt = lvB, where l = length of the moving wire, v = its speed, phi = the magnetic flux linking the circuit, and B = the magnetic field intensity

The motional emf is given by
emf = lvB.

The wire is moved by an external mechanical force.

The number of magnetic field lines inside the loop changes as the area of the circuit changes. I am not assuming a perfect conductor.

MS La Moreaux said:
emf = - d(phi)/dt = lvB
$$EMF = -\frac{d\phi}{dt} = lvB$$

http://labman.phys.utk.edu/phys222core/modules/m5/motional_emf.html

MS La Moreaux said:
Faraday's Law states that the changing magnetic flux linking the circuit results in an emf in the circuit. This would be in addition to the motional emf, since the two principles are independent of each other. The value of the motional emf is equal to the value given by Faraday's Law, and therefore the total emf would be twice the actual value.
Please post the math that shows that there is an issue...

MS La Moreaux said:
The wire is moved by an external mechanical force.
Which does work on the system.

Thank you berkeman for the LaTex and the diagram. I was unable to practice LaTex as described in the LaTex guide.

Physicists seem to have trouble distinguishing math from physics. There is no issue with the math. The issue is with its interpretation. Here we have two identical results from two independent principles, which is no surprise, because the two calculations are identical. It defies logic to suppose that they both apply.

MS La Moreaux said:
It defies logic to suppose that they both apply.

I will send you some tips for posting using LaTeX...

The LaTex guide states that I can practice using the "Preview" button. This button did not work for me. I did not want to post something using LaTex without seeing what it looks like.

Math is not physics. Physics cannot be understood only in math terms. It involves concepts which can be expressed in math, but not understood in terms of math. Two quantities may be equivalent in terms of math, but not equivalent in terms of physics. For example, torque and work have mathematically equivalent units, but they are not physically equivalent.

MS La Moreaux said:
The LaTex guide states that I can practice using the "Preview" button. This button did not work for me. I did not want to post something using LaTex without seeing what it looks like.
This thread is paused temporarily while I try to help the OP learn to use LaTeX...

MS La Moreaux said:
The LaTex guide states that I can practice using the "Preview" button. This button did not work for me. I did not want to post something using LaTex without seeing what it looks like.
I also have this problem. I am not sure if it is a browser issue or something else. However, you can use a website such as https://www.quicklatex.com/ to preview your LaTex before you post it.

berkeman
Drakkith said:
I also have this problem. I am not sure if it is a browser issue or something else. However, you can use a website such as https://www.quicklatex.com/ to preview your LaTex before you post it.
Thanks Drak for the helpful tip. Thread is reopened so the OP can post their work.

Yes, thank you Drakkith.

Let me restate the problem. We have a circuit with an emf. We know the correct value of the emf. But there are two independent, conflicting, explanations for the emf. It seems obvious to me that one of them has to be wrong.

MS La Moreaux said:
Here we have two identical results from two independent principles,
They are not at all independent. How are they conflicting? With brushes the geometry is not fixed and so the integral statement of Faraday.s Law is problematic. One needs to actually understand what one is doing.

How are they conflicting? They are two principles both claiming to be responsible for the emf. That is equivalent to two people claiming to have authored a book, not being coauthors. I certainly believe that they are independent. One is based upon the definition of the magnetic field. I do not know what the other is based upon. I suspect it is nothing. I believe that I understand it completely. I do not understand what the rest of your post is stating. Could you elaborate?

MS La Moreaux said:
How are they conflicting? They are two principles both claiming to be responsible for the emf. That is equivalent to two people claiming to have authored a book, not being coauthors. I certainly believe that they are independent. One is based upon the definition of the magnetic field. I do not know what the other is based upon. I suspect it is nothing. I believe that I understand it completely. I do not understand what the rest of your post is stating. Could you elaborate?
One last chance. Post the math behind your claim of a problem, or this thread is done.

What math? All the math that relates to this case is above. It is not about math. There is no problem with the math. I believe that we should all agree that the math is correct.

MS La Moreaux said:
What math? All the math that relates to this case is above. It is not about math. There is no problem with the math. I believe that we should all agree that the math is correct.
Above where? Send me a PM (private message -- click on my avatar and Start a Conversation) that points to the math ambiguity that you alluding to, and we can reopen this thread. It's closed for now.

## What is Faraday's Law of Induction?

Faraday's Law of Induction states that a change in magnetic flux through a closed loop induces an electromotive force (EMF) in the loop. Mathematically, it is expressed as EMF = -dΦ/dt, where Φ is the magnetic flux. The negative sign indicates the direction of the induced EMF opposes the change in flux, as per Lenz's Law.

## What is motional EMF?

Motional EMF is the electromotive force induced when a conductor moves through a magnetic field. It is given by the formula EMF = Bℓv, where B is the magnetic field strength, ℓ is the length of the conductor, and v is the velocity of the conductor perpendicular to the magnetic field.

## How are Faraday's Law and motional EMF related?

Faraday's Law and motional EMF are related concepts in electromagnetic induction. Faraday's Law describes how a changing magnetic flux induces an EMF, while motional EMF specifically refers to the EMF generated when a conductor moves through a magnetic field. Both phenomena can be described using the concept of changing magnetic flux, but motional EMF is a specific case where the change in flux is due to the physical movement of the conductor.

## Can Faraday's Law and motional EMF be applied simultaneously?

Yes, Faraday's Law and motional EMF can be applied simultaneously in situations where both a changing magnetic field and the motion of a conductor contribute to the induced EMF. In such cases, the total induced EMF is the sum of the EMF due to the changing magnetic flux (Faraday's Law) and the EMF due to the motion of the conductor (motional EMF).

## What are some practical applications of Faraday's Law and motional EMF?

Practical applications of Faraday's Law include transformers, electric generators, and inductors, where changing magnetic fields induce currents. Motional EMF is utilized in devices like railguns, where a moving conductor in a magnetic field generates a current, and in the operation of electric motors, where the motion of conductors through magnetic fields induces EMF to produce mechanical work.

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