Week 3: Potential Energy and Potential 113
the denominator to the numerator (losing the square roots in the process). That is:
rlim≫aV(r, θ) =keq
(r^2 +a^2 − 2 arcos(θ))^1 /^2−
keq
(r^2 +a^2 + 2arcos(θ))^1 /^2= keq
r{
(1− 2 a
rcos(θ) +a2
r^2)−^1 /^2 −(1 + 2a
rcos(θ) +a2
r^2)−^1 /^2
}
=
keq
r{
(1 +
a
rcos(θ)−a^2
2 r^2+...)−(1−
a
rcos(θ)−a^2
2 r^2+...)
}
=
keq
r{
2
a
rcos(θ) +O(
a^3
r^3)}
≈
ke 2 qa
r^2cos(θ)≈kepz
r^2cos(θ) =ke~p·ˆr
r^2
(163)whererˆis a unit vector in the~rdirection. (We used our freedom to rotate the coordinate system
so that~ppoints in an arbitrary direction instead of~zto guess the last result.)
Thisisa very simple form and is a very important one as well! This last equation is the
completely general potential of a point dipoleat a pointP = (r, θ, φ) measured relative to
the dipole center (and withθmeasured from the dipole axis). Note that the answer is azimuthally
symmetric and doesn’t depend onφ, as one expects. Taking the gradient ofthisto find the field
(when you eventually try it) is actually prettyeasy.
We dwell so much on dipoles because they are the most common and important microscopic con-
figuration of charge that produces fields outside of atoms. Atomsare roughly spherically symmetric
and tend to be electrically neutral in isolation. However, atoms are easilypolarizedby any applied
field, including molecular fields. There are molecules (such as the ubiquitous water molecule) that
have permanent electric dipole moments. Speaking as one big bag of (mostly) water to another,
those little electric dipoles can organize in some pretty amazing ways!We will continue to explore
dipole models until we wrap the whole notion up as a macroscopic property of matter called its
dielectric permittivityin the next chapter.
From these two examples it should be simple enough to find the potential at a point due to any
reasonable number of discrete charges provided only that you cando the coordinate geometry needed
to find the distance(s) from the charges to the point of observation. The pythagorean theorem, the
(more general) law of cosines: things like that are thus your best friends in evaluating potentials of
point charges because once you know the distances you just sumkeq/rfor all of those charges.
It’s a bit harder to do a continuous distribution of charge. Let’s lookat a couple of continuous
problems and move on to using the field itself (evaluated with Gauss’s Law) to integrate to the
potential or potential difference.