Magnets — Question 2

Question 2

Rotation work and torque provide independent tests of a magnet moment

A permanent dipole of fixed magnitude m rotates quasistatically in a uniform field B. Positive external work W is needed to turn it from 0 to $120^\circ$.

1. Infer m from W and B using potential energy.

2. Find the magnetic torque magnitude at $120^\circ$ and the work recovered on returning to 0.

Question 2 - Solution

Rotation work and torque provide independent tests of a magnet moment

(a) Method and calculation

$$W=U_f-U_i=-mB\cos120^\circ+mB=\tfrac32mB.$$$$\boxed{m=2W/(3B)}.$$

(b) Interpretation and further calculation

$$|\tau|=mB\sin120^\circ=(2W/3)(\sqrt3/2).$$$$\boxed{|\tau|=W/\sqrt3,\qquad W_{\mathrm{recovered}}=W}.$$

Physical check

Torque and work both have SI unit joule, but torque is a rotational tendency and work is an energy transfer.