Diamagnetism — Question 5

Question 5

A local minimum of field magnitude gives a diamagnetic restoring force

A bead of mass m, volume V, and weak susceptibility chi less than zero is constrained to move along x. Near x=0 the applied field obeys $B^2=B_0^2+ax^2$ with a positive. Ignore gravity along x. Use $U=-\chi VB^2/(2\mu_0)$.

1. Find the restoring force and small-oscillation angular frequency.

2. Find the energy above the minimum at amplitude A and the maximum speed.

Question 5 - Solution

A local minimum of field magnitude gives a diamagnetic restoring force

(a) Method and calculation

$$U-U(0)=\frac{|\chi|Va}{2\mu_0}x^2,\quad\boxed{F_x=-\frac{|\chi|Va}{\mu_0}x}.$$$$m\ddot x+\frac{|\chi|Va}{\mu_0}x=0\Rightarrow\boxed{\omega=\sqrt{\frac{|\chi|Va}{\mu_0m}}}.$$

(b) Interpretation and further calculation

$$\boxed{E_{\mathrm{osc}}=\frac{|\chi|VaA^2}{2\mu_0}}.$$$$\tfrac12mv_{\max}^2=E_{\mathrm{osc}}\Rightarrow\boxed{v_{\max}=\omega A}.$$

Physical check

The result establishes stability only along the constrained coordinate; transverse stability requires the full field geometry.