# Mutual Inductance Equation

## Homework Statement

A toroidal solenoid with mean radius r and cross-sectional area A is wound uniformly with N1 turns. A second toroidal solenoid with N2 turns is wound uniformly on top of the first, so that the two solenoids have the same cross-sectional area and mean radius.
What is the mutual inductance of the two solenoids? Assume that the magnetic field of the first solenoid is uniform across the cross section of the two solenoids.
Express your answer in terms of the variables N1, N2, A, r, magnetic constant μ0 and others appropriate constants.

## The Attempt at a Solution

M = μ0⋅N1⋅N2⋅A this is what I have so far and I know that I need to include the r in the equation somewhere but not really sure where or why. Could somebody please explain. Thank you.

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BvU
Homework Helper
2019 Award
Hi,

Could you show your work ? There seems to be a dimension missing in your M

Hi,

Could you show your work ? There seems to be a dimension missing in your M
I just used the equation $$M = u_0*A*N_1*N_2/l$$ So I assume its the l I'm missing but how does it correspond to r

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gneill
Mentor
I just used the equation $$M = u_0*A*N_1*N_2/l$$ So I assume its the l I'm missing but how does it correspond to r
If the coils were linear then l would be the length of the coils, right? If that length were bent into a circle...

If the coils were linear then l would be the length of the coils, right? If that length were bent into a circle...
I'm not sure

BvU
Homework Helper
2019 Award
Check these coursenotes ( ©2004 Sen-ben Liao, Peter Dourmashkin, and John Belcher, MIT)
In particular (11.2.6) and (11.2.12) . If you confirm for yourself they are equivalent, M follows from (11.2.14)

gneill
Mentor
I'm not sure
If you take a line of length L and bend it to form a circle, can you find the radius of that circle?

If you take a line of length L and bend it to form a circle, can you find the radius of that circle?
Is l not the length of the coil with the turns already in it?

gneill
Mentor
Is l not the length of the coil with the turns already in it?
You're not given the length, you're given the radius r. Your answer should be put in terms of the given variables.

You're not given the length, you're given the radius r. Your answer should be put in terms of the given variables.
Yes, so I need to convert l into r

gneill
Mentor
Yes, so I need to convert l into r
Yes.

Yes.
I don't know how to

gneill
Mentor
If you take a circle of radius r and cut it at one point and straighten it out into a line segment, how long is that line segment?

If you take a circle of radius r and cut it at one point and straighten it out into a line segment, how long is that line segment?
2*pi*r

gneill
Mentor
2*pi*r
Yes!

So l = 2*pi*r

gneill
Mentor
So l = 2*pi*r
Yes.

Yes.
Thank you, got it now