Diamagnetism — Question 2

Question 2

An induced orbital current gives a negative susceptibility in a constrained model

In a constrained classical model an electron of charge -e moves on a fixed circular path of radius r. A slowly changing axial field B induces an azimuthal electric field. Only the change from its initial angular speed is considered; ignore radiation and changes of radius.

1. Use Faraday law and the tangential equation of motion to find the change of angular speed when B grows from zero.

2. Find the induced moment and susceptibility of number density n of identical such loops, using B approximately equal to mu0H.

Question 2 - Solution

An induced orbital current gives a negative susceptibility in a constrained model

(a) Method and calculation

$$2\pi rE_\phi=-\pi r^2\dot B,\quad m_er\,\delta\dot\omega=-eE_\phi=er\dot B/2.$$$$\boxed{\delta\omega=eB/(2m_e)}.$$

(b) Interpretation and further calculation

$$\delta\mu_z=\delta I\pi r^2=-\frac{e\,\delta\omega}{2\pi}\pi r^2=\boxed{-\frac{e^2r^2B}{4m_e}}.$$$$\boxed{\chi=\frac{n\delta\mu_z}{H}\simeq-\frac{\mu_0ne^2r^2}{4m_e}}.$$

Physical check

This explicitly constrained orbit response illustrates induction; it is not a classical thermal-equilibrium theory of atomic magnetism.