2.4 P OTENTIAL DUE TO AN E LECTRIC D IPOLE

. P OTENTIAL DUE TO AN E LECTRIC D IPOLE As we learnt in the last chapter, an electric dipole consists of two charges q and – q separated by a (small) distance a . Its total charge is zero. It is characterised by a dipole moment vector p whose magnitude is q × a and which points in the direction from – q to q (Fig. . ). We also saw that the electric field of a dipole at a point with position vector r depends not just on the magnitude r , but also on the angle between r and p . Further, FIGURE . Variation of potential V with r [in units of ( Q/ πε ) m - ] (blue curve) and field with r [in units of ( Q / πε ) m - ] (black curve) for a point charge Q . the field falls off, at large distance, not as / r (typical of field due to a single charge) but as / r . We, now, determine the electric potential due to a dipole and contrast it with the potential due to a single charge. As before, we take the origin at the centre of the dipole. Now we know that the electric field obeys the superposition principle. Since potential is related to the work done by the field, electrostatic potential also follows the superposition principle. Thus, the potential due to the dipole is the sum of potentials due to the charges q and – q       ( . ) where r and r are the distances of the point P from q and – q , respectively. Now, by geometry, a ar cos θ a ar cos θ ( . ) We take r much greater than a ( a r >> ) and retain terms only upto the first order in a/r cos a a θ       cos a θ   ≅     ( . ) Similarly, cos a θ   ≅     ( . ) Using the Binomial theorem and retaining terms upto the first order in a/r ; we obtain, / cos cos a a θ θ     ≅ ≅         [ . (a)] / cos cos a a θ θ     ≅ ≅         [ . (b)] Using Eqs. ( . ) and ( . ) and p = qa , we get cos cos a p θ θ ( . ) Now, p cos θ = ˆ p r
FIGURE . Quantities involved in the calculation of potential due to a dipole.