Induced current and torque in a rectangular metal loop

[tw12]

Homework Statement

(a) The metallic rectangular loop of length b and width a (ABCD) is mounted in an assembly, which allows it to be rotated about an axis bisecting AD and BC. The plane of the loop makes an angle θ to the horizontal and a uniform magnetic field B is applied vertically upwards (attached is a diagram). Write down an expression for the magnetic flux threading the loop.

(b) The loop is rotated at angular velocity ω such that θ = ωt. Given that the resistance of the wire around the loop is R, find an expression for the induced current in the loop at time t. Hence find the power dissipated in the loop.

(c) By considering the magnetic moment of the loop, find an expression for the torque that must be applied to the loop in order to sustain the rotation at constant angular velocity.

Homework Equations

No equations were given. However, useful equations for the questions might be:

(a) Magnetic flux = ∫B.dA

(b) IR = Change in magnetic flux / Change in time and P = I²R

(c) m = I A and T = IaB(bsinθ) = IABsinθ (where A is the area of the metal loop)

The Attempt at a Solution

(a) Magnetic Flux = ab B cosθ

(b) IR = d (ab cosωt B) / dt

I = (1/R) . d (ab cosωt B) / dt

Subbing this into: P = I²R,

P = (1/R) . (d (ab cosωt B) / dt)²

(c) I wasn't particularly sure where to start on this question, other than this magnetic moment being: m = I ab


Any help on this would be much appreciated, especially part (c), but please correct me on everything else if it's wrong

 

[rude man]

(a) is OK.
(b) you should perform the differentiation and get a closed form expressions for current and power.
(c) (torque) = (mag. moment) x B
work = integrated torque over angle (suggest 360 deg.)
work = avg. power computed in (b) x time. Suggest time of 1 rotation of 360 deg.
equate the two

vectors in bold