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enter image description here

In this attached pictured, the solenoid with the length l and cross sectional area A is solenoid 1 and the other with length say h is solenoid 2.

Why is M₂₁ is equal to M₁₂ as my book claim? M₂₁=N₂Φᴮ²/i₁=μ₀N₁i₁N₂A/li₁=μ₀N₁N₂A/l.

Now, M₁₂=N₁Φᴮ¹/i₂=μ₀N₂i₂N₁A/hi₂=μ₀N₂N₁A/h.

So, given that h≠l, then M₂₁≠M₁₂.

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    \$\begingroup\$ Please make them subscripts for clarity. By definition of the word "mutual", it is a value shared between two inductors that are coupled to some extent and has to be the same. \$\endgroup\$
    – Andy aka
    Commented Dec 28, 2022 at 10:38
  • \$\begingroup\$ Look up reciprocity theorem \$\endgroup\$
    – Tesla23
    Commented Dec 28, 2022 at 10:46
  • \$\begingroup\$ M12 and M21 are, by definition, defined for a "common" flux shared between the 2 inductors. Why should it be "different"? Applying this is perhaps clearer (flux variation): e2 = - M di1/dt or/and e1 = - M di2/dt. electronics.stackexchange.com/questions/579458/… \$\endgroup\$
    – Antonio51
    Commented Dec 28, 2022 at 19:10
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    \$\begingroup\$ Note that M12 and M21 "could" be different if there is a "material" between the two inductors that could be "directional" (let flux pass in one direction, but not in the reverse direction) ... but this is another story. physicsworld.com/a/… \$\endgroup\$
    – Antonio51
    Commented Dec 28, 2022 at 19:17

3 Answers 3

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"Now, M₁₂=N₁Φᴮ¹/i₂=μ₀N₂i₂N₁A/hi₂=μ₀N₂N₁A/h."

This is not true because the effected linkage turns of solenoid 1 are not the total length of l,only h instead.That means the effected turns of solenoid 1 induced by solenoid 2 : $$\frac{N_1}{l} \text{ (is the number of turns per unit length of solenoid 1)} \cdot h $$ then you substitute this effected turns into M₁₂, you get the right answer.

enter image description here

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This video, Mutual inductance of two coaxial solenoids | Electromagnetic induction | Physics | Khan Academy, explains it nicely,

It is a fact, by inspection, that the mutual inductance of any two coils is the same in either direction - unless there is a magnetic field diode in the vicinity.

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  • \$\begingroup\$ The video you linked does not explain why mutual inductances are the same. It just takes advantage of the fact that (in the given conditions) they are the same to simplify the calculation. \$\endgroup\$
    – Sredni Vashtar
    Commented Dec 29, 2022 at 20:06
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This is an example of the reciprocity theorem in electromagnetics.

For mutual inductance proof see: web.mit.edu/sahughes/www/8.022/lec15.pdf section 15.5

It is true in general as long as you don't have non-reciprocal materials.

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