The Magnetic Dipole Moment — Question 5

Question 5

Rigidly combined loops add their moments as vectors in a uniform field

Two small loops are rigidly attached, with magnetic moments $\mu\hat x$ and $\mu\hat y$. A uniform field $B\hat z$ acts on both. Mutual interaction is independent of the rigid assembly orientation and may be omitted.

1. Find the total moment, torque and external-field potential energy.

2. If the second loop is repositioned and fixed so its moment is $-\mu\hat x$, find the new total torque and external-field energy. State the scope of this cancellation.

Question 5 - Solution

Rigidly combined loops add their moments as vectors in a uniform field

(a) Method and calculation

$$\vec\mu_{\mathrm{total}}=\mu(\hat x+\hat y),\quad|\vec\mu_{\mathrm{total}}|=\sqrt2\mu.$$$$\vec\tau=\mu B(\hat x-\hat y),\quad U=-\vec\mu_{\mathrm{total}}\cdot B\hat z=0.$$

(b) Interpretation and further calculation

$$\vec\mu_{\mathrm{total}}\prime=\mu\hat x-\mu\hat x=0.$$$$\vec\tau\prime=0,\quad U\prime=0\quad\text{for the uniform external field}.$$Individual loops can still experience opposing torques and internal mechanical stresses.

Physical check

A vanishing total dipole moment removes the uniform-field dipole coupling; it does not remove all internal forces or all possible gradient effects.