How Do You Calculate Mutual Inductance in Toroidal Solenoids?

[HelpPlease27]

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.

Homework Equations

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.

 

[BvU]

Hi,

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

 

[HelpPlease27]

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

 

[gneill]

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...

 

[HelpPlease27]

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]

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]

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?

 

[HelpPlease27]

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]

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.

 

[HelpPlease27]

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]

Yes, so I need to convert l into r

Yes.

 

[HelpPlease27]

Yes.

I don't know how to

 

[gneill]

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?

 

[HelpPlease27]

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]

2*pi*r

Yes!

 

[HelpPlease27]

So l = 2*pi*r

 

[gneill]

So l = 2*pi*r

Yes.

 

[HelpPlease27]

Yes.

Thank you, got it now