Which method of calculating mutual inductance to use?

Question

I am planning to make a wireless charging device for a school project using coupled induction. However, I have found a few different ways of calculating mutual inductance for coupled solenoids and cannot figure out which to use.

The first method takes the form $M_{12} = \frac{N_2 \Phi_{12}}{i_1}$. For two perfectly coupled solenoids, this simplifies to $N_2 * \frac{B_1 A}{i_1} = N_2 * \frac{\mu_0 N_1 i_1 A/ l_1}{i_1} = \mu_0 \frac{N_1 N_2 A}{l_1}$, where $\Phi_{12}$ is the magnetic flux through solenoid 2 that is generated by solenoid 1, $B_1$ is the magnetic field generated by coil 1, $A$ is the cross-sectional area of the solenoids (it is the same for both), $l_1$ is the length of the first coil, and $N_1$ and $N_2$ are the numbers of loops in each respective solenoid. The assumption of perfect inductive coupling is equivalent to the assumption that all of the magnetic flux generated by coil 1 passes through coil 2 (used for simplicity).

The next method finds $M_{21}$, which is $M_{21} = \frac{N_1 \Phi_{21}}{i_2} = \mu_0 \frac{N_1 N_2 A}{l_2}$, where $\Phi_{21}$ is the magnetic flux through solenoid 1 that is generated by solenoid 2 and $l_2$ is the length of the second coil. The simplification of the formula follows the same steps as for $M_{12}$.

However, the reciprocity theorem states that $M_{12} = M_{21}$. There is a clear discrepancy in the first two calculations of mutual inductance; they vary in the denominator terms $l_1$ and $l_2$. I can't determine the cause of this discrepancy—does reciprocity only apply to solenoids of the same length? I have already examined a derivation provided in "Mutual inductance $M_{12}=M_{21}$: An elementary derivation" as mentioned in this Physics Stack Exchange answer. However, it is still not clear where the discrepancy orignates in my equations for the two mutual inductances.

The last method of calculating mutual inductance that I found was $M = \sqrt{L_1 L_2}$, where $L_1$ and $L_2$ are the inductances of each coil. The derivation of this formula also appears to rely on $M_{12} = M_{21}$, which brings back the same question as before about solenoid lengths.

Any help in figuring out where my calculations may be incorrect (or explaining generally what the mutual inductance of two solenoids depends on) would be highly appreciated. Thank you in advance.

Answer 1

In these types of configurations you must keep in mind that you're approximating fields. Let's say $l_2 > l_1$. Then when calculating $M_{12}$, we need to remember that a part of the 2nd solonoid is going to be extended beyond the shorter 1st solenoid. The field in this outer region is no longer $\mu_0 n_1 I$, and is instead the terrible fringing fields that we want to keep out of calculations.

To amend this, we take advantage of the fact that $M_{12} = M_{21}$. The fields are perhaps too complicated, but whatever they are, their mutual inductances will be the same. However, note that, in theory, if you did do all the calculations, they would agree exactly as you would expect. We're just lazy.