Electric field due to an isolated uniformly Charged Conducting Sphere

In an isolated charged spherical conductor any excess charge on it is distributed uniformly over its outer surface and there is no charge inside it. Hence this problem is the same as that of a charged spherical shell.

Case (i). At an external point

Consider a point P near but outside a uniformly charged sphere of radius R with a charge q. Let σ be the surface density of charge. Then σ = q / (4π R2). P is at a distance r from the centre O. Draw a concentric sphere of radius OP with centre O. This is the Gaussian surface. Let E = electric field at any point on this sphere. At every point E is normal to the surface. The flux through this surface is given by

Ñ„ E . dS = Ñ„ E dS = E (4π r²)

By Gauss’s Law.         E (4π r²) = q/ε_o
    or        E = 1/ (4π ε_o)  q/ r²
    or        E = 1/ (4π ε_o)  q/ r² r

The electric field is, therefore, the same as that due to a charge q situated at the centre of the sphere. Therefore, for points outside the sphere, the charges on the conducting sphere behave as if they were concentrated at the centre of the sphere.

Case (ii). At a point on the surface

    E = ____1____    q
             __ 4π ε_o      R²

Case (iii). At a point inside

Let P’ be an inter point. Though P’ draw concentric sphere. The charge inside this sphere is zero. Hence at all points inside the charged conducting sphere, electric field E = O.