Question

Obtain Gauss law from Coulomb’s law.

Answer 1

Consider a positive point charge Q is surrounded by an imaginary sphere of radius r.
The total electric flux for the entire area is given by,

`Φ_"E" = oint "E" * "d" vec"A" = oint "EdA" cos theta`  ....(1)

The electric field of the point charge is directed radially outward at all points on the surface of the sphere. Therefore the direction of the area element d`vec"A"` is along the electric field `vec"E"` and θ = 0°

`Φ_"E" = "E" oint "dA"` (since cos 0° = 1)

E is a uniform on the surface of the sphere

`Φ_"E" = "E" oint "dA"`   ....(3)

Substituting for `oint`dA = 4πr² and

E = `1/(4pi ε_0) "Q"/"r"^2`, we get

`Φ_"E" = 1/(4pi ε_0) "Q"/"r"^2 xx 4 pi"r"^2`

`= 4pi 1/(4piε_0)`Q

`Φ_"E" = "Q"/ε_0`    .....(4)

This equation is Gauss Law.