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.