The Magnetic Dipole Moment — Question 10

Question 10

Torque and potential energy together determine a dipole magnitude and polar angle

A dipole in a known uniform .200 T field has torque magnitude .00800 newton metre and potential energy -.00600 J. The energy zero is defined by a perpendicular orientation. Let theta lie between zero and pi.

1. Find the dipole moment magnitude and theta without assuming the orientation in advance.

2. Find the quasistatic external work to reverse the moment to its opposite direction, and identify the orientation information still missing.

Question 10 - Solution

Torque and potential energy together determine a dipole magnitude and polar angle

(a) Method and calculation

$$\tau=\mu B\sin\theta,\quad U=-\mu B\cos\theta\Rightarrow(\mu B)^2=\tau^2+U^2.$$$$\mu=\frac{\sqrt{(.00800)^2+(.00600)^2}}{.200}=.0500\,\mathrm{A\,m^2}.$$

(b) Interpretation and further calculation

$$\theta=\tan^{-1}(.00800/.00600)=53.1301^\circ,\quad\cos\theta>0.$$$$\vec\mu\to-\vec\mu:\quad W_{\mathrm{ext}}=U_f-U_i=.00600-(-.00600)=.0120\,\mathrm J.$$The azimuth around B remains undetermined by these two scalar measurements.

Physical check

Energy fixes the sign of the field-parallel component, removing the acute-versus-obtuse ambiguity left by torque magnitude alone.