W9_parallel_resonance.eps

116 Week 3: Potential Energy and Potential

Q

r r

R

S(outer)

S(inner)

Figure 31: A spherical shell of charge of radiusR.

Example 3.4.4: Potential of a Spherical Shell of Charge

Suppose you are given a spherical shell of radiusRof uniformly distributed chargeQ.

Find the field and the potential at all points in space.

If we want to find the potential produced by a spherical shell (or other spherical distribution
of charge) and try to find it by direct integration of the potential of all the charges that make up
the shell, we’ll quickly discover that while it is easy to write down the integral we need to solve in
some system of coordinates, it isn’t so easy todothe integral. It’s still possible – good students of
calculus or students who just want a challenge can tackle it with a reasonable chance of success –
but it isn’t terribly easy. It’s ausefulexample, though, useful enough that I include it in the book
after this “easy way” example, for those very students who wantto give it a try on their own and
then have some way to check or correct their work.

On the other hand, finding theelectric fieldfrom Gauss’s Law isveryeasy (and is done in detail
in Week 2 above, so we won’t repeat the steps here). Try it on your own to make sure that you get:

E~ = 0 (r < R)

E~ = keQ

r^2

ˆr (r > R)

in sphere-centered spherical coordinates. We recall that the potential of any charge distribution with
compact support can be found from the field by directly integratingthe field according to:

V(~r) =−

∫~r

∞

E~·d~l (169)

In this case, we integrate piecewise from the outside in to find the field outside and inside of the
sphere, accordingly. Outside:

V(~r) =−

∫r

∞

keQ

r^2

dr=

keQ

r

(170)

for allr > R. Inside:

V(~r) =−

∫R

∞

keQ

r^2

dr−

∫r

R

0 dr=

keQ

R

(171)

which isconstanteverywhere inside the sphere! This not only makes sense, we’ll makethis into
arule. Any volume where the electrical field vanishes has aconstant potential– we call such a
regionequipotential. We’ll talk about equipotential regions below when discussing conductors in
electrostatic equilibrium (which are, as you can probably already see, equipotential).