Average Torque on an Induced Current Loop

[sengyanglim]

Homework Statement

A rectanglar loop of wire encloses an area A, has a resistance R and a self-inductance L. The loop is suspended from the mid point of one of its sides and placed in a horizontal oscillating magnetic field B = B0 sin ωt such that the field makes an angle θ with the normal to the rectangle. Show that the current in the wire is:
I = −B0 Aω(R^2 + ω^2L^2)^-0.5 cos θcos(ωt − φ)
Find the average torque acting on the rectangle from the magnetic field and
the angles θ when equilibrium is achieved.

Homework Equations

T⃗ =N I A⃗ ×B⃗ =M⃗ ×B⃗

The Attempt at a Solution

I've solved the first part using Ohm's Law and got the expression for the current - I now have an expression for the torque at any point which I think is correct, given by
G= -0.5 B0^2 A^2 ω sin(2θ) cos(ωt − φ) sin(ωt) (R^2 + ω^2L^2)^-0.5
but I need to average this over a cycle - rather than averaging over θ or t alone, I think I need to express θ in terms of t and then average over t, but I don't know how to do this. Any help would be much appreciated!

 

[anathema]

Dear poster,

Thank you for sharing your solution to the first part of the problem. It seems like you are on the right track for finding the average torque, but you are correct in needing to express θ in terms of t in order to properly average over a cycle. One way to do this is to use the relationship between the angle θ and the time t, which can be found by considering the oscillation of the magnetic field B as it changes over time. From there, you can substitute this expression for θ into your torque equation and then average over a cycle of the oscillation. I hope this helps and good luck with the rest of your solution!