# A question regarding the Gauss's Law

• issacnewton
In summary, Gauss's Law for electric fields states that the electric flux through any closed surface is proportional to the enclosed electric charge. This conclusion was reached through the use of the divergence theorem, which states that the volume integral of the divergence of a vector field is equal to the closed surface integral of the vector field normal to the surface. This was independently discovered by Lagrange, Gauss, Green, and Ostrogradsky, with Ostrogradsky providing a rigorous mathematical proof. Gauss used this theorem to formulate his law, which does not rely on the concept of electric field lines.

#### 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 ?

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.

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 .

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.

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

## What is Gauss's Law?

Gauss's Law is a fundamental law in electromagnetism that describes the relationship between electric charges and the electric field they produce. It states that the total electric flux through a closed surface is equal to the enclosed electric charge divided by the permittivity of free space.

## How is Gauss's Law applied?

Gauss's Law is commonly used to calculate the electric field produced by a known distribution of electric charges. It can also be used to determine the total charge enclosed within a given surface by measuring the electric flux through that surface.

## What is the significance of Gauss's Law?

Gauss's Law is significant because it provides a mathematical framework for understanding the behavior of electric charges and their associated electric fields. It is also a fundamental principle in physics and is used in many practical applications, such as designing electrical circuits and devices.

## Are there any limitations to Gauss's Law?

Yes, there are some limitations to Gauss's Law. It only applies to static electric fields and cannot be used to analyze dynamic fields. It also assumes that the electric field is continuous and that the permittivity of free space is constant, which may not always be the case.

## How was Gauss's Law discovered?

Gauss's Law was first formulated by the German mathematician and physicist Carl Friedrich Gauss in the early 19th century. He derived the law using his mathematical skills and experiments with electric charges and fields.