An electron moves on a circular orbit of radius in a plane perpendicular to a uniform…

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The Boundary-Value Problem, Ch Diamagnetism, paramagnetism and magnetic susceptibility

Problem

An electron moves on a circular orbit of radius

r

in a plane perpendicular to a uniform field that is switched on smoothly from

0

B

. ((a)) Using Faraday's law around the orbit, derive the change in orbital speed

\Delta v

and the resulting change in orbital magnetic moment

\Delta m

, and show that the sign of

\Delta m

relative to

\mathbf{B}

is the same whether the electron circulates clockwise or anticlockwise as seen along

\mathbf{B}

. Evaluate

\Delta v

and

\Delta m

for

r = 1.5a_0

(with

a_0 = 5.29\times10^{-11}\,\mathrm{m}

the Bohr radius) and

B = 0.5\,\mathrm{T}

. ((b)) A paramagnetic ion carries a permanent moment of fixed magnitude

m

that can only turn. Starting from

\langle m_z\rangle = mL(mB/k_BT)

with

L(x) = \coth x - 1/x

, derive

\chi_m = \mu_0N_vm^2/3k_BT

in the small-argument limit. Evaluate

\chi_m

for

m = 2\mu_B

N_v = 5\times10^{27}\,\mathrm{m^{-3}}

and

T = 77\,\mathrm{K}

, and check that

mB/k_BT \ll 1

B = 1\,\mathrm{T}

so the linear expansion used is justified.

Related theory

Magnetic materials

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