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www.GOALias.blogspot.comElectric Chargesand FieldsEp=4 π rε 03(r/a >> 1)=−85×10 Cm−12 2 –1 –24 π (8.854 × 10 C N m )×1(15) × 10 m3 −6 3= 1.33 × 10 5 N C –1 .The direction of electric field in this case is opposite to the directionof the dipole moment vector. Again the result agrees with that obtainedbefore.EXAMPLE 1.101.12 DIPOLE IN A UNIFORM EXTERNAL FIELDConsider a permanent dipole of dipole moment p in a uniformexternal field E, as shown in Fig. 1.22. (By permanent dipole, wemean that p exists irrespective of E; it has not been induced by E.)There is a force qE on q and a force –qE on –q. The net force onthe dipole is zero, since E is uniform. However, the charges areseparated, so the forces act at different points, resulting in a torqueon the dipole. When the net force is zero, the torque (couple) isindependent of the origin. Its magnitude equals the magnitude ofeach force multiplied by the arm of the couple (perpendiculardistance between the two antiparallel forces).Magnitude of torque = q E × 2 a sinθ= 2 q a E sinθIts direction is normal to the plane of the paper, coming out of it.The magnitude of p × E is also p E sinθ and its directionis normal to the paper, coming out of it. Thus,τ = p × E (1.22)This torque will tend to align the dipole with the fieldE. When p is aligned with E, the torque is zero.What happens if the field is not uniform? In that case,the net force will evidently be non-zero. In addition therewill, in general, be a torque on the system as before. Thegeneral case is involved, so let us consider the simplersituations when p is parallel to E or antiparallel to E. Ineither case, the net torque is zero, but there is a net forceon the dipole if E is not uniform.Figure 1.23 is self-explanatory. It is easily seen thatwhen p is parallel to E, the dipole has a net force in thedirection of increasing field. When p is antiparallel to E,the net force on the dipole is in the direction of decreasingfield. In general, the force depends on the orientation of pwith respect to E.This brings us to a common observation in frictionalelectricity. A comb run through dry hair attracts pieces ofpaper. The comb, as we know, acquires charge throughfriction. But the paper is not charged. What then explainsthe attractive force? Taking the clue from the precedingFIGURE 1.22 Dipole in auniform electric field.FIGURE 1.23 Electric force on adipole: (a) E parallel to p, (b) Eantiparallel to p.31