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Question

State Gauss law in electrostatics. Using the law derive an expression for electric field due to a uniformly charged thin spherical shell at a point outside the shell.

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Solution

Gauss Law:
Gauss law states that electric flux through any closed surface is equal to the net charge enclosed inside the surface divided by the permittivity of vacuum.
Electric field due to a uniformly charged thin shell:
In the diagram, O is the centre of the spherical shell of radius R, and charge enclosed by the spherical shell is q.
Let a point A is outside the spherical shell, where we have to calculate the electric field.The distance between the centre of spherical shell and the point A.
ϕ=E.Scosθ
So, ϕ=E.4πr2(1)
As per gauss law,
ϕ=qϵo(2)
From equation (1) and (2)
E.4πr2=qϵo
Now,
E=q4πϵor2
This is the electric field at the distance r, outside from the shell.

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