100 Week 2: Continuous Charge and Gauss’s Law
Problem 10.
R
λ
The electric field vanishes inside a uniform spherical shell of charge because the shell has exactly
the right geometry to make the 1/r^2 field produced by opposite sides of the shell cancel according to
the intuition we developed from our derivation of Gauss’s Law. It isn’ta general result for arbitrary
symmetries, however.
Consider aringof charge of radiusRand linear charge densityλ. Pick a pointPthat is in the
plane of the ring but not at the center.
a) Write an expression the field produced by the small pieces of arc subtended by opposed small
angles with vertexP, along the line that bisects this small angle.b) Does this field point towards the nearest arc of the ring or the farthest arc of the ring?
c) Suppose a charge−qis placed at the center of the ring (at equilibrium). Is this equilibrium
stable^39?d) Suppose the electric field dropped off like 1/rinstead of 1/r^2. Would you expect the electric
field to vanish in the plane inside of the ring? Would this be a good form for the electric field
in Edwin Abbot’s novelFlatlandso that they could have a Gauss’s Law too^40?Problem 11.
A uniformly charged nonconducting sphere of radiusais centered on the origin and has a uniform
charge densityρ(r) =ρ 0.
a) Show that at a point within the sphere a distancerfrom the center the electric field is given
by:
E~=ρ^0 ~r
3 ǫ 0=
4 πkρ 0 ~r
3(^39) As a parenthetical aside, note that this is the problem with the ringworld described in Larry Niven’s famous
Ringworldseries of science fiction novels, as gravitational attraction has the same form as the electrostatic attraction
discussed in this problem.
(^40) Alternatively, could a flatlander speculate that reality was really three dimensional because of the apparentfailure
of an expected 1/rforce law? Questions such as this are highly relevant to modern field theorists hoping to infer
extra/hidden dimensions.