W9_parallel_resonance.eps

84 Week 2: Continuous Charge and Gauss’s Law

Now we redo the whole thing for the interior integral:

∮

Sinner

E~·nˆdA = 4πke

∫

V/S

ρdV

Er 4 πr^2 = 4πke

∫r

0

∫ 2 π

0

∫π

0

ρr′^2 sin(θ)dθ dφ dr′

= 4πke(2πρ)

∫r

0

r′^2 dr′

∫ 1

− 1

d(cos(θ))

= 4πke(

4 πr^3

3

ρ) (110)

We divide both sides by 4πr^2 and get:

Er=ke

(

4 πρr

3

)

r < R (111)

This is a common, and important, example – so let’s plot it to make it easier to remember:
Things to note and remember: The field increaseslinearlyinside the sphere and iszeroat the origin,

Er

r

R^2

k Qe

R

Figure 19: Electric field produced by a uniform sphere of charge both inside and outside, as a
function ofr.

not infinite! Outside, the field drops off like 1/r^2 – as you do more and more of these, you’ll come to
expect this to the point where you don’t think twice about it. Any charge distribution with compact
support and a net charge (spherical or not) produces a field thatis dominantlymonopolarand drops
off like 1/r^2 far away from the distribution.

This is very cool! The fact that the field is bounded at the origin meansthat thesingularitythat
appears implicitly in the electrostatic field of apointcharge need not trouble us if the charge isn’t
really apointcharge but is rather a small ball of charge. However, if charge is bound up in a small
finite size ball it producesotherproblems – such as the need for a force to hold it all together, as
electrostatic charge of a single signrepels itself. In the case of a proton, thereissuch a binding force

• the strong nuclear force. In the case of electrons, quarks, elementary particles, thereis(as far as we
  can tell experimentally or predict theoretically) no such force, andhence those particles “should” be,
  and experimentally appear to be, trulypointlike. Which leads to a whole new set of problems (oops,
  that nasty infinity is back and has to be dealt with), the invention of renormalizable quantum field
  theories that soften or throw away the infinity – and in the process, makes physics anenormously
  interesting discipline! Much as wedounderstand at this point, the problem of understanding our
  Universe, especially at the smallest length and time scales, is far fromsolved^36.

(^36) Students who are interested in reading something accessible for the lay person on the subject are encouraged to
pick up a copy ofThe Black Hole War: My Battle with Stephen Hawking to Make theWorld Safe for Quantum
Mechanicsby Leonard Susskind. Great fun, and it will help make many of the concepts discussed here clearer in
context.