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www.GOALias.blogspot.com28PhysicsFIGURE 1.20 Electric field of a dipoleat (a) a point on the axis, (b) a pointon the equatorial plane of the dipole.p is the dipole moment vector ofmagnitude p = q × 2a anddirected from –q to q.The total field at P isq ⎡ 1 1 ⎤E = E ˆ+ q+ E−q= ⎢ −2 2⎥p4 π ε0⎣( r − a) ( r + a)⎦q 4 ar=4 π εo( r − a )For r >> a4qaˆ2 2 2p ˆ(1.14)E =34 πε 0rp (r >> a) (1.15)(ii) For points on the equatorial planeThe magnitudes of the electric fields due to the twocharges +q and –q are given byq 1E+ q =2 24 π ε r + a[1.16(a)]E0q 1= π +[1.16(b)]– q2 24 ε0r aand are equal.The directions of E +qand E –qare as shown inFig. 1.20(b). Clearly, the components normal to the dipoleaxis cancel away. The components along the dipole axisadd up. The total electric field is opposite to ˆp . We haveE = – (E +q+ E –q) cosθ ˆp2qaˆ=− 2 2 3/24 π εo( r + a )p (1.17)At large distances (r >> a), this reduces to2 qaE =− p ˆ ( r a)34 π εor>>(1.18)From Eqs. (1.15) and (1.18), it is clear that the dipole field at largedistances does not involve q and a separately; it depends on the productqa. This suggests the definition of dipole moment. The dipole momentvector p of an electric dipole is defined byp = q × 2a ˆp (1.19)that is, it is a vector whose magnitude is charge q times the separation2a (between the pair of charges q, –q) and the direction is along the linefrom –q to q. In terms of p, the electric field of a dipole at large distancestakes simple forms:At a point on the dipole axis2 pE =34 π ε or(r >> a) (1.20)At a point on the equatorial planepE =−34 π ε or(r >> a) (1.21)