# Average Torque on an Induced Current Loop

1. Dec 15, 2011

### sengyanglim

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
T⃗ =N I A⃗ ×B⃗ =M⃗ ×B⃗

3. 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!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution