Potential Due To A System Of Charges

Consider a system of charges q₁, q₂,…, qn with position vectors r₁, r₂,…, r _n relative to some origin. The potential V₁ at P due to the charge q₁ is

 where r_1P is the distance between q₁  and _P. Similarly, the potential V₂ at P due to q₂ and due to q are given by

where r_2P and r_3P are the distances of P from charges q₂ and q₃, respectively; and so on for the potential due to other charges.

By the superposition principle, the potential V at P due to the total charge configuration is the algebraic sum of the potentials due to the individual charges

The electric field outside the shell is as if the entire charge is concentrated at the centre. Thus, the potential outside the shell is given by

where q is the total charge on the shell and R its radius. The electric field inside the shell is zero. This implies that potential is constant inside the shell (as no work is done in moving a charge inside the shell), and, therefore, equals its value at the surface, which is