Proof of Mutual Induction

[DWigs87]

I am currently a sophomore in college, and as of now, my calculus-based physics class is studying the phenomena of mu tual induction. However, the actual text does not display a proof showing that the proportionallity constant of two coils are equal (Msub(2 1) = Msub(1 2) = M, which is then used in the equation EMF = -M(dIsub(1)/dt) and EMF = -M(dIsub(2)/dt) where I is the steady current in the coil(s)), and the professor did not care to address it, saying it should be assumed.

Does anyone know this how to go about doing the proof for proving that the proportionallity constant is equal for both coils? I have math background up to and including calculus III and differential equations. Any input would be appreciated. Thanks a lot.

 

[Jhero]

Hello,

Thank you for bringing up this question about mutual induction. it is important for us to question and understand the principles and equations that we use in our studies.

To answer your question, the proof for the proportionality constant being equal for both coils is based on the principle of conservation of energy. This principle states that energy cannot be created or destroyed, only transferred from one form to another. In the case of mutual induction, the changing magnetic field in one coil induces an EMF (electromotive force) in the other coil, which in turn creates a current.

Let's start by looking at the equation for mutual induction: EMF = -M(dI/dt), where M is the proportionality constant, I is the current, and t is time. This equation tells us that the EMF induced in a coil is directly proportional to the rate of change of current in the other coil. This means that if the current in one coil changes at a certain rate, the EMF induced in the other coil will also change at the same rate.

Now, let's consider two coils, Coil 1 and Coil 2, with currents I1 and I2 respectively. According to the principle of conservation of energy, the energy transferred from Coil 1 to Coil 2 must be equal to the energy transferred from Coil 2 to Coil 1. In other words, the work done by Coil 1 on Coil 2 must be equal to the work done by Coil 2 on Coil 1.

Using the equation for work (W = Fd), we can rewrite this as:

W1 = F1d1 = -M(dI1/dt)d1
W2 = F2d2 = -M(dI2/dt)d2

Since the proportionality constant M is the same for both coils, we can set these two equations equal to each other:

-M(dI1/dt)d1 = -M(dI2/dt)d2

Solving for M, we get:

M = -(dI1/dt)d1/(dI2/dt)d2

This shows that the proportionality constant M is the same for both coils, as it is independent of the individual coil parameters. Therefore, we can conclude that M1 = M2 = M.

I hope this helps in understanding the proof for the equal proportionality constant in mutual induction. If you have any further questions or would like to