Week 4: Capacitance 129
Problem 9.
Suppose you have a solid sphere with a radiusRand a uniform charge densityρ. Find the potential
at all points in space. Now repeat this for anon-uniform charge density of the formρ(r) =ρ (^0) Rr
(starting by using Gauss’s Law to find the field). Note that this isright on the edgeof being an
“advanced” problem as it requires you to do anintegralto evaluate the total charge inside a Gaussian
surface. To keep it from being “just” an exercise in calculus, note the following:
The volume of a differentially thin spherical shell is its area 4πr′^2 times its thicknessdr′:
dV= 4πr′^2 dr′
The charge in this shell is therefore:
dQ=ρ(r′)4πr′^2 dr′=
4 πρ 0
R
r′^3 dr′
So integrate both sides between sensible limits to find the charge inside a Gaussian sphere of a
given radius inside or outside of the sphere. You can do it! (BTW, I user′instead ofrso you can
makera limit of integration – remember how that works?)
Advanced Problem 10.
Let’s try to use this to understand a little bit about nuclear fission. Suppose that the chargeQin
the previous problem is distributed uniformly in anincompressible fluid. Now imagine that sphere
splitting into two identical, smaller spheres. Find the radiusR′of these two spheres. Obviously,
each sphere has a charge ofQ/2. Find the total electrostatic energy of these two spheres oncethey
have stabilized and are separated by a large distance. Compare theanswer to the answer from the
previous problem. Was energy released? What form would you expect this energy to take?