W9_parallel_resonance.eps

Week 1: Discrete Charge and the Electrostatic Field 33

E~(~r) =ke

∫

ρ(~r 0 )(~r−~r 0 )d^3 r 0

|~r−~r 0 |^3

Because one has to integrate over the vectors, this integral is remarkably difficult. We’ll revisit

it in a much more similar form when we get to electrostaticpotential, a scalar quantity.

• Electric Dipoles
  When two electric charges of equal magnitude and opposite sign arebound together, they form
  anelectric dipole. Thedipole momentof this arrangement is the source of a characteristic
  electrostatic field, thedipole field. The dipole moment of the two charges is defined to be:

~p=q~l

whereqis the magnitude of the charge and~lis the vector that points from the negative charge

to the positive charge.

When an electric dipole~pis placed in auniformelectric fieldE~, the following expressions

describe the force and torque acting on the dipole (which tries to align itself with the applied

field):

F~ = 0

~τ = ~p×E~

Associated with this torque is the following potential energy:

U=−~p·E~

and from this, we can see that the force on the dipole in a more general (non-uniform) field

should be:

F~=−∇~U=∇~(~p·E~)

which is actually nontrivial to compute.

This completes the chapter/week summary. The sections below illuminate these basic facts and
illustrate them with examples.