How Is Displacement Current Calculated in a Parallel Plate Capacitor?

[PenTrik]

Homework Statement

A sinusoidally-varying voltage V(t) = Vosinωt with amplitude Vo = 10 V and frequency of f = ω/(2π) = 100 Hz is impressed across the plates of a circular-shaped parallel plate air-gap capacitor of radius a = 1.0 cm and plate separation d= 0.01 mm. The amplitude of Maxwell's displacement current ID flowing across the gap between the plates of this capacitor is:

Homework Equations

From what I know, the one equation that we have is
I_d=e_0\frac{d\Phi}{dt}
Of course, I'm trying to solve for I_d here

The Attempt at a Solution

I don't really have much of an attempt here. Sort of lost on how to do it.

 

[ideasrule]

It's good to think back to the reason that displacement currents are needed in the first place. See here: http://en.wikipedia.org/wiki/Displacement_current#To_obtain_the_correct_magnetic_field

Ampere's law states that the integral of B*dl across a closed curve is equal to the current through any surface that you can draw around the curve. As the second diagram in that Wikipedia article's section shows, you can draw two different surfaces, one that the wire passes through and one that no current passes through whatsoever. Displacement current is needed to explain this discrepancy, so displacement current must be exactly equal to the current in the wires (or else the discrepancy would remain).

Another way to calculate displacement current would be to use your equation. You can easily calculate the electric field between between the plates as a function of the total charge on the plate and you know the plates' area, so you can calculate electric flux. Differentiating with respect to time would give you d(phi)/dt.