Knoll (knob/bump) on the plate of the capacitor

[sergiokapone]

Homework Statement

The inner surface of one of the plates
flat charged capacitor has a small hemispherical knoll.
Away from it the electric field in the capacitor is equal to ## E_0 ##. Using the principle of superposition find the field at the top and at the base of the knoll.

Homework Equations

Field near metall surface ##E = \frac{\sigma}{\epsilon_0}## (SI units)

The Attempt at a Solution

The idea is to represent plate with knoll as two different things. May be as a plate with hole with the inserted in it sphere, or as a plate with hemisphere lying thereon. Need some help.

 

[theodoros.mihos]

Is this your problem?

 

[rude man]

need picture.
if the knoll protrudes from the plate then the bottom is part of the metalization and the E field there = 0.

 

[sergiokapone]

I found solution. Let's start with the model:

Dipole momentum of the sphere

\begin{equation}
p = \frac{3}{4\pi}VE_0 = r^3 E_0\label{p}.
\end{equation}

Field of the dipole in general
\begin{equation}
\vec E = \frac{3(\vec p\vec r)}{r^5}\vec r - \frac{\vec p}{r^3}. \label{dipE}
\end{equation}

Field of the dipole at the top
\begin{equation}
\vec E_\text{dip} = \frac{2\vec p}{r^3} = 2\vec E_0. \label{dipEup}
\end{equation}

Due to supperposition principle
\begin{equation}
\vec E = \vec E_0 + \vec E_\text{dip} = 3\vec E_0.
\end{equation}

Field of the dipole at the base
\begin{equation}
\vec E = - \frac{\vec p}{r^3}. \label{dipEbase} = - \vec E_0
\end{equation}

Due to supperposition principle
\begin{equation}
\vec E = \vec E_0 + \vec E_\text{dip} = 0.
\end{equation}