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www.GOALias.blogspot.comElectrostatic Potentialand Capacitancewhere ˆr is the unit vector along the position vector OP.The electric potential of a dipole is then given by1 ˆV = pr g24 π ; (r >> a) (2.15)ε0rEquation (2.15) is, as indicated, approximately true only for distanceslarge compared to the size of the dipole, so that higher order terms ina/r are negligible. For a point dipole p at the origin, Eq. (2.15) is, however,exact.From Eq. (2.15), potential on the dipole axis (θ = 0, π ) is given by1 pV =±2(2.16)4 π ε0r(Positive sign for θ = 0, negative sign for θ = π.) The potential in theequatorial plane (θ = π/2) is zero.The important contrasting features of electric potential of a dipolefrom that due to a single charge are clear from Eqs. (2.8) and (2.15):(i) The potential due to a dipole depends not just on r but also on theangle between the position vector r and the dipole moment vector p.(It is, however, axially symmetric about p. That is, if you rotate theposition vector r about p, keeping θ fixed, the points correspondingto P on the cone so generated will have the same potential as at P.)(ii) The electric dipole potential falls off, at large distance, as 1/r 2 , not as1/r, characteristic of the potential due to a single charge. (You canrefer to the Fig. 2.5 for graphs of 1/r 2 versus r and 1/r versus r,drawn there in another context.)2.5 POTENTIAL DUE TO A SYSTEM OF CHARGESConsider a system of charges q 1, q 2,…, q nwith position vectors r 1, r 2,…,r nrelative to some origin (Fig. 2.6). The potential V 1at P due to the chargeq 1is1 q1V1=4 πε0r1Pwhere r 1Pis the distance between q 1and P.Similarly, the potential V 2at P due to q 2andV 3due to q 3are given by1 q2V = 24 π ε r,1 q3V3=4 πε r0 2P0 3Pwhere r 2Pand r 3Pare the distances of P fromcharges q 2and q 3, respectively; and so on for thepotential due to other charges. By thesuperposition principle, the potential V at P dueto the total charge configuration is the algebraicsum of the potentials due to the individualchargesV = V 1+ V 2+ ... + V n(2.17)FIGURE 2.6 Potential at a point due to asystem of charges is the sum of potentialsdue to individual charges.57