2.5 POTENTIAL DUE TO A SYSTEM OF CHARGES

2.6

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

Consider a system of charges q1, q2,…, qn with position vectors r1, r2,…, rn relative to some origin (Fig. 2.6). The potential V1 at P due to the charge q1 is V1=14πε0q1r1P where r1P is the distance between q1 and P.

Similarly, the potential V2 at P due to q2 and V3 due to q3 are given by V2=14πε0q2r2P,V3=14πε0q3r3P

where r2P and r3P 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 (2.17)

V=14πε0(q1r1P+q2r2P++qnrnP) (2.18)

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 Δv and carrying a charge ρΔv. 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 1 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

V=14πε0qr(rR) (2.19(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 2.6) 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=14πε0qR (2.19(b))

Example 2.2

Two charges 3108C and 2108C are located 15cm apart. At what point on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.

2.7
VIEW SOLUTION

Let P be the required point on the x-axis where the potential is zero. If x is the x-coordinate of P, obviously x must be positive. (There is no possibility of potentials due to the two charges adding up to zero for x<0.) If x lies between O and A, we have 14πε0[3×108x×1022×108(15x)×102]=0 where x is in cm. That is, 3x215x=0 which gives x = 9 cm.

If x lies on the extended line OA, the required condition is 3x2x15=0 which gives x=45cm Thus, electric potential is zero at 9 cm and 45 cm away from the positive charge on the side of the negative charge. Note that the formula for potential used in the calculation required choosing potential to be zero at infinity.

Example 2.3

Figures 2.8 (a) and (b) show the field lines of a positive and negative point charge respectively.

2.8

(a)Give the signs of the potential difference VPVQ;VBVA.
(b)Give the sign of the potential energy difference of a small negative charge between the points Q and P; A and B.
(c)Give the sign of the work done by the field in moving a small positive charge from Q to P.
(d)Give the sign of the work done by the external agency in moving a small negative charge from B to A.
(e)Does the kinetic energy of a small negative charge increase or decrease in going from B to A?

VIEW SOLUTION

(a) As V1r, VP>VQ. Thus, (VPVQ) is positive. Also VB is less negative than VA. Thus, VB>VA or (VBVA) is positive.

(b)A small negative charge will be attracted towards positive charge. The negative charge moves from higher potential energy to lower potential energy. Therefore the sign of potential energy difference of a small negative charge between Q and P is positive.
Similarly, (P.E.)A>(P.E.)B and hence sign of potential energy differences is positive.

(c)In moving a small positive charge from Q to P, work has to be done by an external agency against the electric field. Therefore, work done by the field is negative.

(d)In moving a small negative charge from B to A work has to be done by the external agency. It is positive.

(e)Due to force of repulsion on the negative charge, velocity decreases and hence the kinetic energy decreases in going from B to A.

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