Question

The space between two concentric metallic spheres is filled up with a unifrom poorly conducting medium of resistivity `rho` and permittivity `epsilon`. At what moment `t = 0` the inside sphere obtains a certain charge. Find:
(a) the relation between the vectros of displacement current density and conduction current density at an arbitarry point of the medium at the same moment of time,
(b) the displacement current across an arbitrary closed surface wholly located in the medium and enclosing the internal sphere, if at the given moment of time the charge of that sphere is equal to `q`.

Answer 1

(a) There is a radial outward conduction current. Let `Q` be the instantaneous charge on the inner sphere, then,
`j xx 4 pi r^(2) = - (dQ)/(dt)` or, `vec(j) = - (1)/(4pi r^(2)) (d Q)/(dt) hat(r)`
On the other hand `vec(j_(d)) = (del vec(D))/(del t) = (d)/(dt) ((Q)/(4pi r^(2)) hat(r)) = -vec(j)`
(b) At the given moment, `vec(E) = (q)/(4pi epsilon_(0) epsilon r^(2)) hat(r)`
Then, `vec(j_(d)) = (q)/(4pi epsilon_(0) epsilon rho r^(2)) hat(r)`
and `oint vec(j_(a)).d vec(S) = - (q)/(4pi epsilon_(0) epsilon rho) rho (d S cos theta)/(r^(2)) = - (q)/(epsilon epsilon_(0) rho)`
The surface intergal must be `-ve` because `vec(j_(d))` being opposite of `vec(j)`, is inward.