PHY 3323 November 3, 2010 Exam #2

PHY 3323
November 3, 2010
Exam #2
What happens to you here is forever. Understand that in advance. We shall crush
you down to the point from which there is no coming back. Things will happen to
you from which you could not recover, if you lived a thousand years. . . . You will
be hollow. We shall squeeze you empty, and then we shall fill you with ourselves.
(1) Two infinite parallel grounded conducting planes are held a distance a apart. A point
charge q is placed in the region between them, a distance x from one plate. This
problem concerns the force (which is a vector!) on q.
a) What form does dimensional analysis imply for the force? (10 points)
b) What must be the limiting form of the force for x ≪ a? (10 points)
c) What must be the limiting form of the force for x = a/2? (10 points)
d) Use the method of images to express the force as an infinite sum. (5 points)
(2) A conducting sphere of radius a, at potential V0 , is surrounded by a thin concentric
spherical shell of radius b, over which someone has glued a surface charge
σ(θ) = k cos(θ) ,
where k is a constant, and θ is the usual spherical coordinate.
a) What is the potential in each of the three regions: 0 < r < a, a < r < b and
b < r < ∞? (20 points)
b) What is the induced surface charge σi (θ) on the conductor? (10 points)
c) What are the charge and the electric dipole moment (which is a vector!) of this
system? (10 points)
(3) Consider two half-infinite slabs of linear isotropic dielectric which join at the plane
z = 0. The top slab has permitivity ǫ1 and the bottom slab has permitivity ǫ2 . There
are no free charges at any finite point, however, the top slab has a constant electric
~ 1 = E0 sin(α)b
x − cos(α)b
z ,
where E0 and α are constants.
a) What are the electric displacement vector and the polarization in the top slab?
(15 points)
b) What is the electric field vector in the bottom slab? (20 points)
c) What is the bound surface charge density at the interface? (15 points)