Magnets — Question 5

Question 5

A field gradient translates a magnet even at zero torque

A small permanent dipole is held parallel to +x on the symmetry axis of an external field whose on-axis magnitude is $B(x)=B_0+ax+bx^2$, positive in the region of interest. Its fixed moment is $m\hat x$.

1. Find its potential energy, axial force, and torque on the axis.

2. Find the axial equilibrium and its stability; obtain the frequency for mass M when stable.

Question 5 - Solution

A field gradient translates a magnet even at zero torque

(a) Method and calculation

$$U(x)=-m(B_0+ax+bx^2),\qquad\boxed{F_x=m(a+2bx)}.$$$$\boxed{\boldsymbol\tau=\mathbf m\times\mathbf B=0}.$$

(b) Interpretation and further calculation

$$x_*=-a/(2b),\qquad U^{\prime\prime}=-2mb>0\iff b<0.$$$$\boxed{\omega_x=\sqrt{-2mb/M}}\quad(b<0).$$

Physical check

This is constrained one-dimensional stability; it does not establish stability against rotation or transverse motion.