# Mutual inductance of a solenoid wrapped around part of a toroid

WJSwanson
*This isn't a problem so much as needing to fill in the gaps from a lecture where I was taking notes on painkillers due to surgery. The problem statement is my best guess as to what exactly was said by the instructor.

## Homework Statement

"Find the mutual inductance (M) of a square-bore toroid that has a solenoid wrapped partially around it. Assume the toroid and solenoid behave as their respective ideals. No numerical values aside from physical constants are included; we are looking for an analytical solution."

Inner and outer radii of the square-bore toroid: a and b, respectively.
Edge length the toroid bore: h
Number of turns in the solenoid and the toroid: Ns and Nt, respectively.

Now here's the work I had in my notes:
1.
$EMF_s = -M \frac{di_t}{dt}$

$EMF_t = -M \frac{di_s}{dt}$

2.
$N_t \frac{d\Phi_{B,s}}{dt} = M \frac{di_t}{dt}$

$N_s \frac{d\Phi_{B,t}}{dt} = M \frac{di_s}{dt}$

3.
$\oint \vec{B} . d\vec{s} = \mu_0 N_t i_t$

4.
$\Phi_{B,1} = \int \vec{B} . d\vec{A} = \int^b_a \frac{\mu_0 N_2 i_2 h}{2\pi r} dr = \frac{\mu_0 N i_t h}{2\pi} ln(\frac{b}{a})$

5.
$N_s \Phi_{B,1} = M i_2 \Rightarrow M = \frac{N_s N_t \mu_0 h}{2\pi} ln(\frac{b}{a})$

## Homework Equations

$EMF = -d\Phi_B / dt$

EMF vs inductance:
$EMF = -L di/dt$

Ampere's Law:
$\oint \vec{B} . d\vec{s} = \mu_0 i_{encl}$

Self-inductance of a solenoid:
$L = N^2_s \mu_0 A * \frac{1}{l}$

Self-inductance of a toroid:
$L = \int^b_a N^2_t \mu_0 h dr / 2\pi r = \frac{N^2_t \mu_0 h}{2\pi} \int^b_a \frac{dr}{r} = \frac{N^2_t \mu_0 h}{2\pi} ln(\frac{b}{a})$

## The Attempt at a Solution

So the question arises, did I write everything down correctly? That might not be the case. Assuming I did, I run into some weirdness.

Using Faraday's Law, we find that
$EMF_s = -N_s \frac{d \Phi_{B,s}}{dt} \Rightarrow N_s \frac{d\Phi_{B,s}}{dt} = M \frac{di_t}{dt}$

and

$EMF_t = -N_t \frac{d \Phi_{B,t}}{dt} \Rightarrow N_t \frac{d\Phi_{B,t}}{dt} = M \frac{di_t}{dt}$.

So now we have:
$N_s \frac{d\Phi_{B,s}}{dt} = M \frac{di_t}{dt}$
&
$N_t \frac{d\Phi_{B,t}}{dt} = M \frac{di_s}{dt}$.

From there, I'm not sure what to do. I suppose it would be time to pick which geometry has the easier flux calculation (probably the toroid, since we aren't given the dimensions of the solenoid) to find di/dt for its counterpart?

For the toroid:

$\oint \vec{B} . d\vec{s} = \mu_0 i_{encl.} = \mu_0 N_t i_t$
$d(\oint \vec{B} . d\vec{s}) / dt = \mu_0 \frac{d i_{encl.}}{dt} = \frac{\mu_0}{L} \frac{d\Phi_B}{dt}$

or at least I think. I really am not sure where any of the rest of this stuff comes from, though. I would seriously appreciate anyone who can help me figure this out. :)

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