Module 2 : Electrostatics
Lecture 10 : Poisson Equations
 Poisson Equation
  Differential form of Gauss's law,
\begin{displaymath}\vec\nabla\cdot\vec E = \frac{\rho}{\epsilon_0}\end{displaymath}
  Using $\vec E = -\nabla\phi$,
 
\begin{displaymath}\vec\nabla\vec E = -\nabla\cdot(\nabla\phi) = - \nabla^2\phi\end{displaymath}
  so that
 
\begin{displaymath}\nabla^2\phi = -\frac{\rho}{\epsilon_0}\end{displaymath}
  This is Poisson equation. In cartesian form,
 
  A formal soltion to Poisson equation can be written down by using the property Dirac - function discussed earlier. It can be seen that
 
   
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