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Date: 28-7-2016
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Charge Distribution in Inhomogeneous Medium
A stationary current distribution is established in a medium that is isotropic but not necessarily homogeneous. Show that the medium will in general acquire a volume distribution of charge whose density is (in Gaussian units)
where σ and ε are the conductivity and the dielectric permittivity of the medium and ϕ is the potential.
SOLUTION
From the Maxwell equation for D = εE
we have
(1)
Substituting J/σ for E in (S.3.20.1) (for an isotropic medium where J is the current density), we find
From the condition that the current is stationary and substituting back for J,
and so
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