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

  1. Jun 6, 2015 #1
    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. jcsd
  3. Jun 6, 2015 #2
    Is this your problem?
     
  4. Jun 6, 2015 #3

    rude man

    User Avatar
    Homework Helper
    Gold Member

    need picture.
    if the knoll protrudes from the plate then the bottom is part of the metalization and the E field there = 0.
     
  5. Jun 6, 2015 #4
    I found solution. Let's start with the model:
    403a07446e9060338a7f7a080574c4c4.png

    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}
     
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