Electrostatic Potential At A Point Due To Group Of Electric Charges

Consider a system of charges q1, q2,…, qn with position vectors r1, r2,…,r n relative to some origin (Fig.). 

The potential V1 at P due to the charge q1 is,

V1 = q1 / 4 π ε₀ r (1P)

where r(1P) is the distance between q1 and P.

Fig. Potential at a point due to a system of charges is the sum of potentials due to individual charges.

Similarly, the potential V2 at P due to q2 and V3 due to q3 are given by, 

V2 = q2 / 4 π ε₀ r (2P)   

V3 = q3 / 4 π ε₀ r (3P)

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

V = V1 + V2 + ... + Vn

V=  1/4 π ε_{0 [}q1/ r (1P)+ q2/ r (2P)+q3/ r (3P)+....+qn/ r (nP)]

Note :

For a uniformly charged spherical shell, 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

V = q / 4 π ε₀ r    (r>= R)

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,

V = q / 4 π ε₀ R