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DIPOLES II We now consider the effect of fields on dipoles. The situation is as shown in the following drawing. In general, the field produces both a torque and a force. Torque R F q E r q E r 2 2 2 2 p p E r E r 2 2 2 2 Now let 0 . Then pE r Force F q E r q E r 2 2 Hence if E r E r 2 2 Then F = 0. In other words, a dipole feels no net force in a uniform field. In a non-uniform field there will be a force. To find it we consider a special case in which the dipole is parallel to the field Then E q E r q E r xˆ 2 2 If E is due to another dipole, p 2 p 2 xˆ located at the origin then 2kp 2 E xˆ R3 and 1 1 2kqp 2 1 1 F q 2kp 2 xˆ 3 3 3 3 r3 r 1 r 1 2r 2 2 2r Now (1 + x)n =1 + nx + ….. 2kqp 2 2kqp 2 3 6kp1p2 F 1 3 1 3 xˆ 2r 2r r 3 r3 r r4 Since τ ~1/r3 and F ~ 1/r4 it is normally torque that dominates. Potential Energy of Dipole in Electric Field Consider assembling the dipole from infinity. PE = PEi + work I do to move charges from infinity to their final positions First move both to the center of the dipole. Since the charges are equal and opposite, this takes no work. Now move them at right angles to the field to a distance ℓ apart. This also takes no work because F qE is perpendicular to the distance moved. At this point we have Now rotate by an angle to This requires work w qE sin q E sin 2 2 q E sin q E cos pE cos p E PE p E Torque on One Dipole Due to Another k k p2 E p 2 3 p1 rˆ rˆ p1 3 p1 rˆ p 2 rˆ p 2 p1 r3 r3 Suppose p1 p1xˆ and p 2 p 2 yˆ . Then 2kp1p 2 k 3p1 p2zˆ p1p 2zˆ zˆ r3 r3 Hence dipole 2 will rotate counter clockwise, as seen from above page, until it lines up with dipole 1.