- Problem 1 [10 points]
Using Dirac delta functions in the appropriate coordinate system, express the following charge distributions as three dimensional charge densities ρ(r).
a) In spherical coordinates, a charge Q uniformly distributed over an infinitesmally thin spherical shell of radius R.
b) In cylindrical coordinates, a charge λ per unit length uniformly distributed over an infinitely long cylindrical surface of radius b.
c) In cylindrical coordinates, a charge Q spread uniformly over a flat circular disk of negligible thickness and radius R, centered in the xy plane at z=0.
d) The same as (c), but using spherical coordinates.
- Problem 2 [10 points]
The time averaged electrostatic potential of a neutral hydrogen atom is given by
where r=|r| is the radial distance, q is the magnitude of the electron charge, and α-1=a0/2 with a0 the Bohr radius. Find the distribution of charge (both continuous and discrete) which will give rise to this potential, and interpret your result physically.
- Problem 3 [10 points]
Consider a plane polarized electromagnetic wave described by the vector and scalar potentials,
A(r, t) = Aoei(k·r-ωt), φ(r, t) = φoei(k·r-ωt)
where the orientation of the vector Ao is arbitrary.
a) Using Maxwell's equations, find the relationship that must hold between Ao and φo.
b) Using the principal of gauge invariance, show that one can transform to a new but physically equivalent vector potential which is transversely polarized, i.e. Ao·k=0. Explicitly find the gauge transformation function χ that does this.
c) What is the scalar potential φ(r, t) in the gauge of part (b)?
- Problem 4 [10 points]
Prove this mean value theorem: For charge-free space in the electrostatic limit, the value of the electrostatic potential φ at any point in space is equal to the average of the potential over the surface of any sphere centered on that point.
Hint: Use the fact that where there are no charges ∇2φ=0. Functions which satisfy this Laplace's equation are called harmonic functions; harmonic functions obey the above mean value theorem.
- Problem 5 [10 points]
a) In lecture we solved the problem of the electric field from a spherical shell of radius R with uniform surface charge density σ=q/(4πR2). Consider now the problem where this shell is of finite thickness d. That is, there is a uniform charge density ρ in a spherical shell of finite thickness from radius R to radius R+d, such that the total charge on this shell is q. Find the potential φ(r) by solving Poisson's equation (there may be easier ways to do it, but do it this way!), then take the gradient to get E(r). Sketch φ(r) and E(r) vs r. Now take the limit d→0 keeping ρd=σ constant. Compare your result with the case of the infinitesmally thin shell done in lecture.
b) Consider an infinitesmally thin spherical shell of radius R with a total charge q uniformly distributed over its surface, and a concentric infinitesmally thin spherical shell of radius R+d with total charge -q uniformly distributed over its surface. Find the potential φ(r) by solving Poisson's equation for this geometry, then take the gradient to get E(r). Sketch φ(r) and E(r) vs r. Now take the limit d→0 keeping qd constant. What do you find? This is the limit of an infinitesmally thin dipole layer.
- Problem 6 [10 points]
Prove Green's reciprocation theorm: If φ is the potential due to a volume charge density ρ within a volume V and a surface charge density σ on the conducting surface S bounding the volume V, while φ' is the potential for the same geometry but for a different ρ' and σ', then
∫Vd3r ρφ' + ∫Sda σφ' = ∫Vd3r ρ'φ + ∫Sda σ'φ
Hint: Consider Green's 2nd identity.
- Problem 7 [10 points]
Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use Green's reciprocation theorem to prove that the total induced charge on one of the planes is equal to -q times the fractional perpendicular distance of the point charge from the other plane. (Hint: As your comparison electrostatic problem with the same surfaces choose one whose charge densities and potential are known and simple.)