## Related questions with answers

A parallel-plate capacitor is made using two circular plates of radius a, with the bottom plate on the xy plane, centered at the origin. The top plate is located at z = d, with its center on the z axis. Potential $V_0$ is on the top plate; the bottom plate is grounded. Dielectric having radially dependent permittivity fills the region between plates. The permittivity is given by $\epsilon(\rho)=\epsilon_{0}\left(1+\rho^{2} / a^{2}\right)$ Find (a)V(z); (b) E; (c) Q; (d)C.

Solution

Verified$\textbf{a)}$

We have learned in the Problem 6.36. that even if the permittivity varies in space, the Poisson equation is still valid if permittivity varies in a direction perpendicular to the electric field. That is the case here, so we can start with the Laplace equation (since $\rho_v=0$):

$\begin{gather*} \nabla^2V=0\implies \dfrac{\partial^2V}{\partial z^2}=0 \end{gather*}$

The solution that satifies the boundary conditions is a simple linear function:

$\begin{gather*} \boxed{V(z)=V_0\dfrac{z}{d}} \end{gather*}$

$\textbf{b)}$

The electric field is the negative gradient of the potential:

$\begin{gather*} \mathbf E=-\nabla V=-\boxed{\dfrac{V_0}{d}\mathbf a_z} \end{gather*}$

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