Electricity and Magnetism I (P331) Homework 5 Due: Thursday , October 15, 4:00 PM 1. (a) Show that the electric ﬁeld of a pure dipole can be written in a coordinate-free form as E ( r ) = 1 4 π±0 1 r 3 [3( p · ˆ r )ˆ r-p ] . (1) (b) Show that the energy of an ideal dipole p in an electric ﬁeld E is given by U =-p · E . (2) To do this consider the amount of work done to move a dipole from inﬁnity through an electric ﬁeld and then orient the dipole in some speciﬁc direction with respect to the electric ﬁeld. (c) Use your answers to (a) and (b) to show that the interaction energy of two dipoles separated by a distance r is U = 1 4 π±0 1 r 3 [ p 1 · p 2-3( p 1 · ˆ r )( p 2 · ˆ r )] . (3) 2. Imagine three charged rods with length ` and charge Q. The rods are arranged so that one end of each rod is at the origin and the other end is a distance ` away from the origin along the x , y , and z axis. (a) Find the dipole moment of this charge charge distribution
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