W9_parallel_resonance.eps

Week 1: Discrete Charge and the Electrostatic Field 57

Problem 7.

+Q

−q

+q

l

r

A positive point charge +Qis at the origin, and a dipole of moment~pis at a distanceraway
and pointing in the radial direction (wherer≫L, the physical length of the dipole) as shown.

a) Show that the force exerted on the dipole by the point charge is attractive and has a magnitude

Fr≈

2 kQp

r^3.

b) Now assume that the dipole is centered at the origin and that a point chargeQis a distancer

along the line of the dipole. Using Newton’s third law and your result forpart (a), show that

at the location of the positive point charge the electric field due to the dipole is toward the

dipole and has a magnitude of

Er≈

2 kp

r^3

.

Again, 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. Or,
you might be able to do it by considering this one a special case of the previous problem, if you can
mentally rotate coordinate systems...