82 Week 2: Continuous Charge and Gauss’s Law
every time, as when you know something well enough to be slightly bored writing it out, that’s just
about perfect, isn’t it?
Again we can count up the charge insideS 2 on the thumbs of one hand. It is the total charge
on the shell! Which is, in fact (noting thatdAfor a spherical shell of radiusaisa^2 sin(θ)dθ dφ):
QS =
∫
Sσ 0 dA=∫ 2 π0dφ∫π0sinθdθ a^2 σ 0 = 2πa^2 σ 0∫ 1
− 1d(cosθ)= 4πa^2 σ 0 (105)which wecouldhave done using our heads instead of calculus, but there is a clever trick in this
example (using sinθdθ=−d(cosθ) to change variables and limits on theθintegral) which we used
above when explicitly integrating above and which we’ll have occasion to use again in other problems.
Finally, we write out Gauss’s law and solve forEr:φe=Er(4πr^2 ) =QS
ǫ 0(106)
or
Er=Qs
4 πǫ 0 r^2=
keQs
r^2(107)
where once again Gauss’s law gets us extremely simply something we probably should remember
from last semester, which is thatthe field of a spherically symmetric charge distribution outside that
distributionis the same as that of apoint charge with the same net charge located the origin.
This isexactly what we got the hard way earlier in this chapter! The hard way being
an explicit (and quite difficult) integral over the actual charge distribution. The fact that we get the
same answer should give us some confidence that Gauss’s Law is trueand correct. It also convinces
us that when we can use it it ismuch easierthan explicit integration!
In lecture your instructor will probably do a few more difficult problems – perhaps a solid sphere
of charge, or multiple spherical shells, or even a solid sphere with a charge distribution likeρ(r) =Ar
whereAis a constant! You should be able to doanyproblem with a spherical distribution of charge
that you can integrate or sum inside any given Gaussian sphere usingthis method.
Also note that once one has done asinglespherical shell, one can easily do as many concentric
shells as you might have on your fingers and toes using thesuperposition principle. Simply add the
field produced by each shell at the point in question (which might be inside or outside the given
shell) to that produced by all the other shells! There’s a homework problem to help you learn that
- do it!
Example 2.3.2: Electric Field of a Solid Sphere of Charge
Find the electric field at all points in space of a solid insulating sphere with uniform
charge densityρand radiusRJust for grins, let’s do a teensy bit of your homework together. Note well that youdon’t get
to just copy this onto your paper! In order tolearnthis and get it right three weeks from now on
an exam, you have to be able to do itwithoutlooking, or copying. So by all means, go through
the example, study it, figure it out, then close this book or put asideyour digital interface, get out
paper, anddo it on your own without looking– as many times as necessary to make the steps, and
reasoning,easyto you. Go over it in multiple passes, work on it in your groups, review itin your
notes (your teacher/professor probably did this example in class), discuss it in recitation.Learnit.