Question

State and prove Gauss’s theorem.

Answer 1

Hint: In this question firstly, we will learn about the Gauss Theorem and the relationship that it states between electric flux and the charge, and then we will understand about the theorem with the help of definition and examples. And using this knowledge, we will be able to approach our answer.

Step-By-Step answer:
Friedrich Gauss in 1835 related the electric field on the closed surface and net charge which was enclosed by that surface. Gauss law is also known as the Gauss’s flux theorem which is the law related to electric charge distribution resulting from the electric field.
When an area of the surface projected in a plane perpendicular to the field is multiplied by the electric field is the electric flux.
It states that, the total electric flux of a given surface is equal to the $\dfrac{1}{{E\theta }}$ times of the total charge enclosed in it or amount of charge contained within that surface.

Let us consider a point charge ‘q’ and an electric field observed of radius ‘r’.
Now, we will take an area as ds (the direction of area vector) and an electric field on the surface as ‘E’.
de = E (vector) ds (vector) $\cos \theta $
$\theta = \int {\overrightarrow {E.dA} } = \int {\overrightarrow {E.dA.} } $$\cos \theta $
$\theta = 0$
Cos 0 = 1
$\theta = \int {\overrightarrow {E.dA} } $ =$\overrightarrow E $. $4\pi {r^2}$
The electric field at any point ‘P’ is
$\overrightarrow E $ = $\dfrac{1}{{4\pi e\theta }}$ .$\dfrac{q}{{{r^2}}}$
Pointing the field radially outward at point ‘P’
$\theta = \dfrac{q}{{e\theta }}$
Hence Proved!

Although it is proved for spherical surfaces, yet it holds good for any arbitrary surface.
Gauss Theorem gives a relationship between the total flux which passes through a closed surface and the net charge enclosed within the surface.

Note - Friedrich Gauss in 1835 related the electric field on the closed surface but it was published in 1867. Gauss law is about the relationship of electric charge and electric field. For an electric field, it describes the static electric field generated due to the distribution of electric charges.