A solid glass sphere of relative permittivity is placed in a uniform applied electric…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter · Magnetic materials

The Boundary-Value Problem, Ch Magnetisation, bound currents and H, completed

Problem

A solid glass sphere of relative permittivity

\varepsilon_r = 6

is placed in a uniform applied electric field

E_0 = 1000\,\mathrm{V\,m^{-1}}

, far from any other charge. A solid nickel-alloy sphere of relative permeability

\mu_r = 600

is placed in a uniform applied magnetic field

B_0 = 10\,\mathrm{mT}

, far from any free current. Both spheres are linear, isotropic and free of any permanent polarisation or magnetisation. ((a)) Using the correspondence between the dielectric sphere and the linear magnetisable sphere, calculate the interior field

E_{\rm in}

of the glass sphere and the interior fields

B_{\rm in}

and

H_{\rm in}

of the nickel-alloy sphere. ((b)) Both problems rest on solving Laplace's equation for a scalar potential (

V

for the glass sphere,

\Phi_M

for the nickel-alloy sphere) matched at

r=a

by continuity of the potential together with continuity of the normal component of

\mathbf{D}

for the glass and of

\mathbf{B}

for the nickel alloy; the jump left in the normal component of

\mathbf{E}

or of

\mathbf{H}

is set by the surface term

\mathbf{P}\cdot\hat{\mathbf{n}}

\mathbf{M}\cdot\hat{\mathbf{n}}

respectively. Explain, using an order-of-magnitude comparison of the numbers just found together with the fact that permanent magnets can hold

M \sim 10^6\,\mathrm{A\,m^{-1}}

while stable electrets are rare and weakly polarised, why the indeterminacy of

\mathbf{H}

(it is not fixed by free current alone) matters far more in laboratory practice than the parallel indeterminacy of

\mathbf{D}

(it is not fixed by free charge alone).

Related theory

Magnetic materials

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