Inductor

Greetings. I used the Biot-Savart Law to get the magnetic flux density (B) on the square loop axis, at an arbitrary distance from the source loop. I then added up the contributions to the peak solenoid voltage from each circular coil, keeping in mind that the B value is different for each coil. The contribution from each coil is (A coil * omega * B max), where B max depends on I max and on geometry. I ended up getting 26.424 mV peak (18.685 mV RMS), and I'm wondering why this method went wrong. I assumed that the flux density at the center of each circular loop was representative of the flux density for the entire loop. Would that cause the error?

Here is the proof for the symmetry of the mutual inductance.

Consider $M_{12}=\phi_1/I_2$ , where $\phi_1$ is the magnetic flux in loop #1, generated by the current $I_2$ in loop #2. The magnetic field by the current $I_2$ can be calculated by the Biot Savart law, but here it is more practical to use the vector potential. The relationship between the magnetic field and the vector potential $\vec A$ is $\vec B=\vec \nabla \times \vec A$ . The vector potential can be expressed with the current as

$A(\vec r)=\frac {\mu_0 I_2}{4 \pi} \oint_{C_2} \frac{\vec{d r_2}}{|\vec r-\vec r_2|}$

where $\vec r_2$ is on loop #2.

The flux in loop #1 is $\phi_1 = \int_{S_1} \vec B \vec{dS}$ . Using the vector potential and Stokes's theorem we can write this as

$\phi_1 = \int_{S_1} (\vec \nabla \times \vec A) \vec{dS}= \oint_{C_1} \vec A \vec{dr_1}$

Where the integration is over the first loop. Combining the two results we get

$\phi_1 = \frac {\mu_0 I_2}{4 \pi} \oint_{C_1} \oint_{C_2} \frac{\vec{d r_1}\vec{dr_2}}{|\vec r_1-\vec r_2|}$

and

$M_{12} = \frac {\mu_0 }{4 \pi} \oint_{C_1} \oint_{C_2} \frac{\vec{d r_1}\vec{dr_2}}{|\vec r_1-\vec r_2|}$

This expression is symmetric with respect to interchanging the indices 1 and 2.