Force between two current-carrying loops

[fogvajarash]

Homework Statement

Consider two circular, parallel, coaxial loops which are almost in contact. They are separated by 1.20mm, have 13.6cm as radius each. Both loops carry a current of I = 127A but in opposite directions. Find the magnetic force that the top loop exerts on the bottom loop.

Homework Equations

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The Attempt at a Solution

I tried to "readapt" the force between two parallel wires, to be able to use it for two loops, in which we would have $F = BIl = 2\pi BIr$. Then, I used the formula for the field that a loop exerts on its axis, but now I'm doubting if this is true as actually this field is not felt by the other current loop. Now I am stuck on which way to calculate the force that one loop exerts on the other. I was thinking of finding the force exerted on one small piece of the circle dr and then try to integrate in order to find the total force.

Thank you for your time and patience.

 

[rude man]

Remember magnetic moment? Think of coil 1 as producing a B field along its axis, which you can easily compute. Then, think of coil 2 as having a magnetic moment, and then associate a potential energy U of that moment at the distance between the coils. So U = U(x) where x is the distance between the coils.
Then, appeal to the principle of virtual work which says that force x Δ(distance) = Δ(potential energy).
This problem is a bit advanced for an introductory course but not at all unmanageable (no elliptic integrals, legendre polynomials, etc.).