Mutual Inductance - Two rectangular loops

[lukeb87]

Homework Statement

Consider two rectangular loops. An outer and inner loop. Assign outer loop a length a and width b. Assign inner loop length c and width d.

Apply a current to the outer loop.

Determine an expression for the mutual inductance between the loops.

Homework Equations

V2 = M di1/dt
B(r) = (mu0*I) / 2*Pi*r
magnetic flux through loop = ((mu0*L*I)/(2*Pi)) * ln(b/a)
EMF = -d(flux)/dt

The Attempt at a Solution

Here is my attempt and a visual representation of the problem:

http://i910.photobucket.com/albums/ac301/lukebaldan/assign_attempt1.jpg"

We know that mutual inductance can be determined via:

V2 = M di1/dt

My methodology is to rework the simple case consisting of a long infinite line and a rectangular loop. The solution is trivial (Double integral over area element involving the definition of the magnetic field outside of wire).

In this case we can consider the superposition of each side of the outer loop. The current is identical and the adjacent lines are at 90 degrees so we ignore them when considering the other side of the loop.

Therefore the contribution from the top and bottom lines is simply twice that of the simple single line case.

This is repeated for the sides.

Is my solution correct? I cannot think of another way to solve the problem. My attempt seems logical to me. Can anyone point me in the correct direction? Any help is much appreciated.

Thanks,
Luke

 

[praharmitra]

You should seriously clean up your notation! took me quite a bit of time to figure it out.

A and B are outer loop lengths. a and b are inner loop lengths (not c and d).

In your calculations, you have used variables a and b to define some other quantity even though a and b have already been used! Make sure you don't mess up because of that.

But other than that, the answer is right!

note that in the final answer

a = (B+'b')/2
b = (B-'b')/2
c = (A+'a')/2
d = (A-'a')/2

where 'a' and 'b' are the original variables as described in the problem.

 

[lukeb87]

Hi there. Sorry i apologise. I meant to describe the problem in an easy fashion, My drawing is a much better representation of the problem.

Thanks for the acknowledgment.