Mutual Inductance of Coaxial Coils: Is M12 Always Equal to M21?

In summary, the conversation discusses the relationship between mutual inductance values for two coaxial coils, with one being shorter and having a smaller area. It is determined that M21 always equals M12, and can be shown using an energy argument. The expression for M21 is computed, taking into account the number of turns and magnetic permeability. The conversation also touches on the difficulty in deriving M12, as it is uncertain how much of the flux from coil 2 couples into coil 1. However, it is ultimately concluded that M12 must equal M21.
  • #1
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Homework Statement



Not sure if I'm correct or wrong..because usually M12 = M21

The Attempt at a Solution



Here coil 1 and coil 2 are coaxial, with coil 2 being shorter and having smaller area.

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  • #2
M21 always = M12. That can be shown by an energy argument.

It's correct I think to say that
M21 = N2phi1/i1. That computes to
M21 = N2A2μn1
where N2 is the number of turns in the inside coil 2. This agrees with your expression of M12 (I think you reverse the conventional sequence of the subscripts, but OK).

This is because the flux inside coil 2 is clearly very nearly the same as that inside coil 1.

It's more difficult to argue in the same manner to derive M12 because the question is how much of the flux generated by coil 2 really couples into coil 1.

The answer however has to be such that M12 = M21. I have looked at several derivations of mutual inductance of various coil configurations and in each case they pick the easier computation for Mij and then assume (correctly) that Mji = Mij.
 

FAQ: Mutual Inductance of Coaxial Coils: Is M12 Always Equal to M21?

1. What is mutual inductance?

Mutual inductance is a phenomenon that occurs when two coils are placed near each other, causing a change in the magnetic field of one coil to induce a voltage in the other coil.

2. How does mutual inductance of coaxial coils work?

The mutual inductance of coaxial coils is based on the principle of electromagnetic induction. When an alternating current flows through one coil, it creates a changing magnetic field. This changing magnetic field then induces an alternating current in the other coil, causing a voltage to be produced.

3. What factors affect the mutual inductance of coaxial coils?

The mutual inductance of coaxial coils is affected by the number of turns in each coil, the distance between the coils, and the permeability of the material between the coils. The shape and orientation of the coils also play a role in determining the mutual inductance.

4. How is mutual inductance of coaxial coils calculated?

The mutual inductance of coaxial coils can be calculated using the equation M = μ0 * N1 * N2 * A / l, where μ0 is the permeability of free space, N1 and N2 are the number of turns in each coil, A is the area of the coils, and l is the distance between the coils.

5. What are some practical applications of mutual inductance of coaxial coils?

Mutual inductance of coaxial coils has various applications in electrical engineering, such as in transformers, motors, and generators. It is also used in wireless power transfer systems, where the mutual inductance between two coils is used to transfer energy without the need for physical contact.

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