# Can Coulomb's law be used to derive Gauss's law? How?

##### 1 Answer

I shall outline a better method. Not just by assuming the surface is spherical with area

So, to start with I shall prove it for the field due to a single point charge. (At O)

Since superposition principle applies to electric fields, this can be generalised to any number of charges quite easily.

There it is, given below.

#### Explanation:

Consider a charged particle

Take an arbitrary surface element

Let the normal on

Thus,

Now, by Coulomb's law,

Which gives,

Now,

Thus, the flux through

Integrating over the entire surface,

But, the net solid angle subtended by a closed surface at an internal point is always

Thus,

Which is Gauss law in integral form.

One may derive the differential form from Coulomb's law as well by taking the Divergence of the electric field and then proceeding with vector calculus methods.

The differential form looks something like this,