How to Derive Displacement Current and Energy Density in a Capacitor?

[hitman0097]

Homework Statement

Show that the displacement current density which flows when a parallel plate capacitor gets charged is ε(dE/dt) where ε is the permittivity of the medium between the capacitor. What is the energy density stored in the capacitor and the force per unit area (pressure) on the plates?

Homework Equations

D=ε_oE + p= ε_oE +ε_oχE=εE
p=\alphaE (mircoscopic)
p=χE (marcoscopic)
(Constitute of Relations) p is the polarization of the medium

The Attempt at a Solution

(a) I am really confused as to how to show the first part, which I think is needed for the whole problem.
(b) For the work, from my notes I have\tau\tau=\sigmaE which means it behaves equivallently to Ohms Law V=IR so the Voltage which is the work done per unit charge is equal to that expression.
(c) For the energy density of the capacitor U_e=1/2 ε_o E^2
And the Total energy density is U_t=ε_o/2 *E^2 +B^2/2μ_o which reduced to 2ε_oE^2
(d) P=1/2 ε_oE^2 +B^2/μ
P=ε_o E^2
P=s/c (s is the Poynting vector)

I am just really confused but this is what my notes and external sites have led me to. Any help to make sense of this is appreciated (I'll be talking to my professor today too as well so hopefully I can understand this better)

 

[hitman0097]

tau is the displacement current, and sigma should be the conductivity? I think

 

[hitman0097]

I didn't need the equation for D in the first part, All I needed was C=eA/d for capactiance and q=CV

 

[hitman0097]

So If i take the time derivative of q=cv I get the current I = e A/d *dV/dt

 

[hitman0097]

Nvm, I figured most of it now.

 

[vela]

Oh, I thought you had figured it out five hours ago.