9.0 Gauss's law

  Electrostatics

Gauss's law states that the total electric flux through an imaginary closed surface is proportional to the total electric charge enclosed within the surface.

In other words, the net electric flux through any closed surface is equal to the net charge inside the surface divided by ${\varepsilon _0}$.

Mathematically it can be written as, $${\phi _e} = \frac{{{q_{in}}}}{{{\varepsilon _0}}}$$ or $$\oint\limits_S {\overrightarrow E .\,d\overrightarrow S } = \frac{{{q_{in}}}}{{{\varepsilon _0}}}$$ If the area of the enclosed surface is known then the above equation is reduced as, $$ES = \frac{{{q_{in}}}}{{{\varepsilon _0}}}$$
where,

${q_{in}}:$ Net charge inside the surface
$\overrightarrow E :$ Electric field at any point on the surface
${\phi _e}:$ Electric flux through any closed surface
$S:$ Area of the enclosed surface where electric field is perpendicular to the surface.

Note: This technique is useful for calculating the electric field in situations where a degree of symmetry is high.

Some important aspects of application of Gauss's theorem for determination of electric field

1. The imaginary surface which is chosen for application of Gauss's law is called a Gaussian surface.
   Any Gaussian surface can be chosen but it should not pass through a discrete charge. However, it can pass through a continuous charge distribution.
   
   
2. Gauss's theorem is mostly used for symmetric charge distribution.
   
   
3. The term $q_{in}$ corresponds to sum of all charges enclosed within the gaussian surface.
   
   
4. Gauss's theorem is based on the inverse square dependence on distance contained in the Coulomb's law. Any violation of Gauss's theorem will indicate the departure from the inverse square law.