A point charge `+q` is held at a distance R from the centre of an uncharged spherical conductor of radius ` a(a lt R)`. Calculate the potential of the conductor in the presence of q.

Explanation

Remember that the superposition principle is valid enen in the presence of conductors.

'q' alone would produce an electric field at all points within the volume of the conductor . The conductor must some how cancel the field due to q at all poits interior to it. but there is no charge given to the conductor . the conductor solves this poroblem in a remarkable way. it pulls out positive and negative charges of equal magnitude from its volume and spreads them on its. surface in such a manner that at every point in its interior the field due to q is cancelled by the field due to the surface charges. the potential `phi` within the sphere is uniform , because it is a conducting sphere. if we can find the value of `phi` at any piont in the sphere, our problem is solved. All the points within the sphere, there is one point which is best suited for the the same distance 'a' from the centre of the sphere and therefore, the potential at C due . to all the surface charges (irrespective of their distribution ) is
`oint (sigma."ds")/(4 pi epsilon_(0)a) = (1)/(4 pi epsilon_(0)a) oint sigma`.ds = 0
Because the total charge on the surface of the sphere is zero in this problem . so only contribution to potential at C comes from q now, and this is equal to `(q)/(4 pi epsilon_(0) R)`
Thus the potential of the sphere is `(q)/(4 pi epsilon_(0)R)`
Note that the above discussion can be extended to the case where the net charge on the spherical conductor has any given value Q. Several point charges outside the conductor can also be taken into account.