# Gauss's Law

The relationship between a general distribution of electric charge and its field is given by Gauss' Law. It can be used to calculate the distribution of charge over a system of electric charges. It is useful because in certain cases, it is easier to use than Coulomb's Law.

We have seen how a charges create electric fields.

$\nabla . \vec E = \frac{\rho}{\epsilon_0}$

In integral form

$\int \vec E . d\vec A = \frac{q}{\epsilon_0}$

It works by defining an imaginary closed surface around a distribution of charges and then calculating the how much of the flux cuts through our imaginary surface.

The closed surface can be chosen depending on the geometry of the problem and so we can use symmetry to make the problem to simplify the integration.

## The normal vector

We define a normal vector to the surface of the imaginary surface and its direction is determined by convention. When the flux and the normal are in the same direction the magnitude is positive and when the flux and the normal vector are in opposite directions the magnitude is negative.

Also when the normal vector points away from the charge it encloses.

### Deriving Coulomb's Law from Gauss' Law

Consider a charge q. Apply Gauss' Law by surrounding the charge by a spherical shell centered on the charge.

$\int \vec E. d\vec A = \frac{q}{\epsilon_0}$

By the symmetry, the electric field will be a constant, therefore the integral just reverts to the surface area of the sphere which is 4&pi r^2.

Or $\vec E(\vec r) = \frac{q }{4\pi \epsilon_0 r^2}\hat r$