Module 2 : Electrostatics
Lecture 7 : GAUSS'S LAW
 
\begin{displaymath}\oint \vec E\cdot \vec {dS} = \frac{Q_{enclosed}}{\epsilon_0}\end{displaymath}

The law is valid for arbitry shaped surface, real or imaginary.

Its physical content is the same as that of Coulomb's law.
In practice, it allows evaluation of electric field in many practical situations by forming imagined surfaces
 

which exploit symmetry of the problem. Such surfaces are called Gaussian surfaces .

GAUSS'S LAW - Differential form
  The integral form of Gauss's law can be converted to a differential form by using the divergence theorem. If $V$is the volume enclosed by the surface S,
 
\begin{displaymath}\oint_S\vec E\cdot\vec{dS} = \int_V\vec\nabla\cdot\vec E dv \eqno(A)\end{displaymath}
  If $\rho$is the volume charge density,
 
\begin{displaymath}Q = \int_V\rho dv\eqno(B)\end{displaymath}
  Thus we have
 
\begin{displaymath}\nabla\cdot\vec E = \frac{\rho}{\epsilon_0}\end{displaymath}
   
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