Description
Problems
1. Spherical cavity: Consider a sphere of radius R that has a hollow spherical cavity of
radius b inside it. The center of the big sphere is at the origin, the center of the cavity
is at ~a. The volume of the big sphere (excluding the cavity) is uniformly charged with
a charge density ρ0.
(a) (25 pts) Derive the electric field (both magnitude and direction) at an arbitrary
point inside the cavity. (Find a compact expression in terms of the given parameters.)
2. Hydrogen atom, Jackson 1.5 (25 pts): The time-averaged potential of a neutral
hydrogen atom is given by
Φ = q
4πε0
e
−αr
r
1 +
αr
2
(1)
where q is the magnitude of electronic charge, and α
−1 = a0/2, a0 being the Bohr
radius. Find the distribution of charge (both continuous and discrete) that will give
this potential and interpret your result physically.
3. Field of a thin disc: An infinitely thin round disk of radius R has its symmetry axis
on the z-axis. It is uniformly charged with total charge q.
(a) (10 pts) Write an expression for the charge density of the disk ρ(~r) using appropriate coordinate variables.
(b) (5 pts) Determine the cartesian surface density σ(x, y) (from your expression for
ρ(~r)).
(c) (10 pts) Calculate by direct integration the electric field E~ (~r) at an arbitrary
point on the z-axis (from your expression for ρ(~r 0
)).
(d) (5 pts) Find the limits of the field for z R and for z R and explain the
results.
4. Equipotential surface (20 pts): Two opposite point charges q1 and −q2 are positioned distance d apart. (Here q1 and q2 are unequal positive numbers.) Show that the
equipotential surfaces in this system include a sphere of finite radius. Find the location
of the center of the sphere and its radius. What is the value of the potential on the
surface of this sphere ? (We use such a normalization that the value of the potential
at infinity is zero.)