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www.GOALias.blogspot.comElectrostatic Potentialand CapacitanceEquation (2.8) is true for anysign of the charge Q, though weconsidered Q > 0 in its derivation.For Q < 0, V < 0, i.e., work done (bythe external force) per unit positivetest charge in bringing it frominfinity to the point is negative. Thisis equivalent to saying that workdone by the electrostatic force inbringing the unit positive chargeform infinity to the point P ispositive. [This is as it should be,since for Q < 0, the force on a unitpositive test charge is attractive, sothat the electrostatic force and thedisplacement (from infinity to P) are FIGURE 2.4 Variation of potential V with r [in units ofin the same direction.] Finally, we (Q/4πε 0) m -1 ] (blue curve) and field with r [in unitsof (Q/4πεnote that Eq. (2.8) is consistent with0) m -2 ] (black curve) for a point charge Q.the choice that potential at infinitybe zero.Figure (2.4) shows how the electrostatic potential ( ∝ 1/r) and theelectrostatic field ( ∝ 1/r 2 ) varies with r.Example 2.1(a) Calculate the potential at a point P due to a charge of 4 × 10 –7 Clocated 9 cm away.(b) Hence obtain the work done in bringing a charge of 2 × 10 –9 Cfrom infinity to the point P. Does the answer depend on the pathalong which the charge is brought?Solution−71 Q9 2 –2 4×10 C(a) V = = 9× 10 Nm C ×4πε0r0.09m= 4 × 10 4 V−9 4(b) W = qV = 2× 10 C× 4×10 V= 8 × 10 –5 JNo, work done will be path independent. Any arbitrary infinitesimalpath can be resolved into two perpendicular displacements: One alongr and another perpendicular to r. The work done corresponding tothe later will be zero.EXAMPLE 2.12.4 POTENTIAL DUE TO AN ELECTRIC DIPOLEAs we learnt in the last chapter, an electric dipole consists of two chargesq and –q separated by a (small) distance 2a. Its total charge is zero. It ischaracterised by a dipole moment vector p whose magnitude is q × 2aand which points in the direction from –q to q (Fig. 2.5). We also saw thatthe electric field of a dipole at a point with position vector r depends notjust on the magnitude r, but also on the angle between r and p. Further,55