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