A thin conducting spherical shell of radius R has charge Q spread uniformly over its surface. Using Gauss’s law, derive an expression for an electric field at a point outside the shell.

#### Solution

According to Gauss law,

`epsi_0EointdA=q`

Where,

*q* is the point charge

*E* is electric field due to the point charge

*d**A* is a small area on the Gaussian surface at any distance and

`epsi_0` is the proportionality constant

For a spherical shell at distance *r* from the point charge, the integral `ointdA` is merely the sum of all differential of *dA* on the sphere.

Therefore, `oint dA =4pir^2`

`epsi_0E(4pir^2) = q`

or `,E = q/(epsi4pir^2)`

Therefore, for a thin conducting spherical shell of radius *R* and charge *Q*, spread uniformly over its surface, the electric field at any point outside the shell is

`E = Q/(e_0 4pir^2)`

Where r is the distance of the point from the centre of the shell.

`E = q/(4piepsi_0r^2)`