Module 2 : Electrostatics
Lecture 11 : Conductors and Dielectric
  The potential due to the whole volume is
\begin{displaymath}\phi = \frac{1}{4\pi\epsilon_0}\int_{volume}\frac{\vec P\cdot... ...4\pi\epsilon_0}\int_{volume}\vec P\cdot\nabla(\frac{1}{r})d\tau\end{displaymath}
  where, we have used
 
\begin{displaymath}\nabla(\frac{1}{r}) = \frac{\hat r}{r^2}\end{displaymath}
  Use the vector identity
 
\begin{displaymath}\vec\nabla\cdot(\vec A f(r)) = \vec A\cdot\nabla f(r) + f(r)\vec\nabla\cdot\vec A\end{displaymath}
  Substituting $\vec A = \vec P$and $f(r)=1/r$,
 
\begin{displaymath}\vec\nabla\cdot(\frac{\vec P}{ r}) = \vec P\cdot\nabla(\frac{1}{r})+ \frac{1}{r}\vec\nabla\cdot\vec P \end{displaymath}
  we get
 
\begin{displaymath}\phi = \frac{1}{4\pi\epsilon_o}\int_{vol}\vec\nabla\cdot(\fra... ...i\epsilon_o} \int_{vol} \frac{1}{r}\vec\nabla\cdot\vec P d\tau \end{displaymath}
   
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