Sea water at frequency `v=4 xx 10^(8)` Hz has permittivity `epsilon ~~ 80 epsilon_(0)` permeability `mu= mu_(0)` and resistivity `rho= 0.25`M. Imagine a parallel plate capacitor immersed in sea water and driven by an alternating voltage source `V(t) =V_(0) " sin "(2pivt).` What fraction of the conduction current density is the displacement current density ?

Explanation

Let a be the distance between the plates. The applied voltage is given by `V(t)=V_0sin 2pivt`.
The applied electric field, `E=(V(t))/d=(V_0)/d sin 2pivt`
The conduction current density, `J_C=E/rho =(V_0)/(rhod) sin 2pivt=J_(0C) sin 2piv t`
where `(V_0)/(rhod)=J_(0C)`=max. conduction current density.
Displacement current density is given by
`J_D=(I_D)/A=(in_0in_r d(phi_E)//dt)/A=(in_0 in_rd(EA))/(Adt)=in_0in_r (dE)/(dt) =in_0in_rd/(dt)((V_0)/d sin 2pivt)`
`=in_0in_r (V_0)/d 2piv cos 2pivt =J_(0D) cos 2pivt`
Where `in_0in_r (V_0)/dxx2piv=J_(0D)`=max. displacement current desity
`:. (J_(0D))/(J_(0C))=(in_0 in_r V_0 2piv//d)/(V_0//(rhod))=2piv in_0in_r rho=4pi in_0 v in_r rho//2=1/(9xx10^9)xx(4xx10^8)xx80xx0.25//2=4/9`