Physics 350: Daily Exercise
The following questions questions refer to the material
you were to read in preparation for class. Questions one and two require you to
write a three or four sentence response.
1.
"Earnshaw's
Theorem" states that it is impossible to hold a point charge in stable
equilibrium by electrostatic forces alone. Which of the following statements
justifies this theorem?

a) The electric force is conservative.

b) Solutions to Laplace's equation have no local maxima or minima.

c) Solutions to Laplace's equation for a given set of boundary
conditions are unique.

d) A charge in stable equilibrium would violate the second law of
thermodynamics.
1) a and b, especially b. in order to be in a stable equilibrium, the energy
of the charge must be at a minimum, but Laplace's solutions lack minima, they
have saddles!
Answers: 1) Statement be justifies the theorem. If a charge was held static
by only electrostatic forces, then there would be a local minimum in the
potential. By Laplace's equation, this cannot be true
Answers: 1. b) Solutions to Laplace's equation have no local maxima or
minima.
Answers: 1.b (i'm not sure on this one)
Answers: 1.b) Solutions to Laplace's equation have no local maxima or
minima.
Answers: 1) the answer is b): solutions to laplace's equation have no local
maxima or minima. but in this case, since the problem refers to stable
equilibrium, and not to unstable equilibrium, a better statement would simply be
that "solutions to laplace's equation have no local minima". wow, i
hope i'm right.
Answers: 1. I'm really not sure which of those statements justifies that 
but it makes sense to me. Even if you could hold the first charge static with
other static charges, you wouldn't be able to keep the others static. Once they
start moving, the first one wont be in equilibrium anymore. This probably has to
do with the lack of extrema in solutions to Laplace's equation ...