# A question regarding the Gauss's Law

issacnewton
Hi

I have some questions about the Gauss's law for electric field. Wikipedia definition says

The electric flux through any closed surface is proportional to the enclosed electric charge

Now how did Mr Gauss arrive at this conclusion regarding any closed surface ?
That is a very general statement. And during his time, were there instruments which could have lead to the formulation of such general principles ?

And can you please explain this without "electric field lines" since they are not
physically real lines ?

## Answers and Replies

Jasso
If i remember correctly, it's a direct result of the divergence theorem.

Essentially, if there is a vector field A, then the volume integral of the divergence of A in a volume V is equal to the closed surface integral of A normal to the surface enclosing V:

$$\int (\nabla \cdot A) dV = \oint (A \cdot n) dS$$

In Gauss's Law, the vector field is the electric field.

issacnewton
But, Jasso, I think the equation you quote is a later mathematical formulation of the Gauss's Law. Gauss's must have arrived at it from some experimental evidence .

Jasso
The divergence theorem was independantly discovered by Lagrange, Gauss, Green, and Ostrogradsky, With Ostrogradsky providing a rigorous mathematical proof of it. It's a pure mathematics with physical applications.

issacnewton
ok, That makes sense then. So Gauss just used this theorem to come up with the Law..