# Homework Help: Knoll (knob/bump) on the plate of the capacitor

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1. Jun 6, 2015

### sergiokapone

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
Field near metall surface $E = \frac{\sigma}{\epsilon_0}$ (SI units)

3. 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.

2. Jun 6, 2015

3. Jun 6, 2015

### 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.

4. Jun 6, 2015

### sergiokapone

Dipole momentum of the sphere

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

Field of the dipole in general

\vec E = \frac{3(\vec p\vec r)}{r^5}\vec r - \frac{\vec p}{r^3}. \label{dipE}

Field of the dipole at the top

\vec E_\text{dip} = \frac{2\vec p}{r^3} = 2\vec E_0. \label{dipEup}

Due to supperposition principle

\vec E = \vec E_0 + \vec E_\text{dip} = 3\vec E_0.

Field of the dipole at the base

\vec E = - \frac{\vec p}{r^3}. \label{dipEbase} = - \vec E_0

Due to supperposition principle

\vec E = \vec E_0 + \vec E_\text{dip} = 0.