An isolated parallel plate capacitor has circular plates of radius `4.0cm`. If the gap is filled with a partially conducting material of dielectric constant `K` and conductivity `5.0xx10^-14Omega^-1m^-1`. When the capacitor is charged to a surface charge density of `15muC//cm^2`, the initial current between the plates is `1.0muA`?
a. Determine the value of dielectric constant `K`.
b. If the total joule heating produced is `7500J`, determine the separation of the capacitor plates.

Explanation

a. This is basically a problem of discharging of a capacitor from inside the capacitor. Charge at any time `t` is
Here, `q_0=` (area of plates )(surface charge density)
and discharging current,
`i=((-dq)/(dt))=q_0/tau_C.e^(-t//tau_C)=i_0e^(-t//tau_C)`
Here `i_0=q_0/tau_C=q_0/(CR)`
`C=(Kepsilon_0A)/d` and `R=d/(sigmaA)`
`:. CR=(Kepsilon_0)/sigma`
Therfore, `i_0=q_0/((Kepsilon_0)/sigma)=(sigmaq_0)/(Kepsilon_0)impliesK.(sigmaq_0)/(i_0epsilon_0)`
substituting the values, we have
`K=((5.0xx10^-14)(pi)(4.0)^2(15xx10^-6))/((1.0xx10^-6)(8.86xx10^-12))=4.25`
b. `U=1/2q_0^2/C=1/2q_0^2/((Kepsilon_0A)/d)`
`:. d=(2Kepsilon_0AU)/q_0^2`
`=(2xx4.25xx8.86xx10^-12xxpixx(4.0xx10^-2)^2xx7500)/((15xx10^-6xxpixx4.0xx4.0)^2)`
`=5.0xx10^-3m=5.0mm`