. P OTENTIAL DUE TO A S YSTEM OF C HARGES Consider a system of charges q , q ,…, q n with position vectors r , r ,…, r n relative to some origin (Fig. . ).
The potential V at P due to the charge q is 1P where r 1P is the distance between q and P. Similarly, the potential V at P due to q and V due to q are given by 2P , 3P where r 2P and r 3P are the distances of P from charges q and q , 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 = V + V + ... + V n ( .
) FIGURE . Potential at a point due to a system of charges is the sum of potentials due to individual charges. E XAMPLE . 1P 2P P ......
n n ( . ) If we have a continuous charge distribution characterised by a charge density ρ ( r ), we divide it, as before, into small volume elements each of size ∆ v and carrying a charge ρ∆ v. We then determine the potential due to each volume element and sum (strictly speaking , integrate) over all such contributions, and thus determine the potential due to the entire distribution. We have seen in Chapter that 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 ( ≥ [ . (a)] where q is the total charge on the shell and R its radius. The electric field inside the shell is zero. This implies (Section .
) that potential is constant inside the shell (as no work is done in moving a charge inside