W9_parallel_resonance.eps

56 Week 1: Discrete Charge and the Electrostatic Field

Problem 6.

x 1

−q +q

y

E = Cxx

An electric dipole consists of two charges +qand−qseparated by a very small distance 2a. Its
center is on thexaxis atx=x 1 , and it points along thexaxis in the positivexdirection. The
dipole is in a nonuniform electric field which is also in thexdirection, given byE~=Cxxˆ, whereC
is a constant.

a) Find (write down, it’s trivial)~p, the (vector) dipole moment of this electric dipole. Note its

magnitudepx.

b) Find the force on the positive charge and that on the negative charge, and show that the net

force on the dipole isC pxxˆ.

c) Show thatin general, if a dipole lies along thexaxis in an electric field in thexdirection so

that~p=pxxˆ, the net force on the dipole is given approximately by:

Fx=

dEx

dx

px

where the derivative of the field is evaluated at the position of the dipole.

You will probably need to use a Binomial/Taylor expansion to deal with the “r≫L” condition.
Your instructor or TA will help you with this if you have no idea how to proceed.