Electrodynamics I Midterm Exam - Part A - Closed Book KSU 2014/10/23
Name
Instructions: Use SI units. Where appropriate, define all variables or symbols you use, in words. Try
to tell about the physics involved, more than the mathematics, if possible.
1. (3) Write Gauss’ Law in differential form. Explain the physical meaning.
2. (3) Write an expression that gives the electrostatic field energy in vacuum.
3. (3) Show how to get the capacitance of an isolated spherical conductor of radius R. How
large in µF is the capacitance of the Earth (R = 6380 km), considered as a large conductor?
4. (3) Write a differential equation that a Green function G(~r, ~r 0 ) for Poisson’s equation must
satisfy, for Dirichlet boundary conditions.
5. (3) A problem has boundaries with Dirichlet boundary conditions. How do you write the
solution to the Poisson equation for electrostatic potential Φ(~r ) using a Green’s function?
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6. (3) Give a condition (possibly as an inequality) that identifies the limit where classical E&M
theory should be replaced by quantum theory. Explain it.
7. (3) A charge density ρ(~r ) is invariant when the system is rotated through any angle around
the z-axis. How can you write the general solution of Poisson’s equation for the potential
Φ(~r ) in this situation?
8. (3) A linear and isotropic dielectric medium has electric susceptibility χ. How does χ enter
in the formulas for the electric polarization and the electric permittivity?
9. (3) Give a formula that determines the electric dipole moment of an arbitrary but localized
charge density ρ(~r ).
10. (3) If a point electric dipole p~ is located at position ~r0 , what electrostatic potential does it
produce at an arbitrary position ~r ?
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11. (6) For the point dipole of the previous question, what electric field does it produce at an
arbitrary position ~r?
12. Use delta-functions to express the charge density ρ(~r ) for the following charge distributions,
in the indicated coordinate systems:
a) (3) A charge Q distributed uniformly over an infinitely thin circular ring of radius a
centered on the z-axis and lying in the plane z = b. Use spherical coordinates (r, θ, φ).
b) (3) A point charge q on the x-axis at x = x0 . Use cylindrical coordinates (ρ, φ, z).
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13. A point charge q is placed at a distance d > a from the center of an uncharged isolated metal
sphere of radius a.
a) (6) Determine the electric force acting on q due to the sphere, for arbitrary d > a. Is it
attractive or repulsive? Explain.
b) (4) Find the asymptotic force law for d a.
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Electrodynamics I Midterm Exam 1 - Part B - Open Book KSU 2014/10/23
Name
Instructions: Use SI units. Please show the details of your derivations here. Explain your reasoning for
full credit. Open-book only, no notes.
1. An infinitely thin ring of total charge Q has a radius a, and is placed centered on the z-axis
in the plane z = b, above a grounded infinite plane conductor at z = 0. The plane of the
ring is parallel to the plane of the conductor.
a) (8) Find the electric potential Φ(z) along the z-axis anywhere z > 0, which is the axis
of the ring.
√
b) (8) Expand your result 2 2
√ of part a in power series, one valid for z < a + b , and another
series valid for z > a2 + b2 .
c) (8) Use the result of part b to find the electric potential Φ(r, θ, φ) for any points above
the grounded plane.
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2. A very long conducting cylinder with a circular cross section of radius a is placed with its
axis a distance b > a away from and parallel to a grounded plane conductor. The cylinder
is held at fixed potential V relative to the grounded plane.
a) (10) Use the method of images and show that by an appropriate choice of image line
charges, the equipotentials are circles. Hint: The image line charge within the cylinder
does not need to be along its axis.
b) (8) Find how the circle center and radius of an equipotential circle depend on a chosen
value of potential Φ between 0 and V .
c) (6) Calculate the capacitance per unit length of the cylinder/plane system.
d) (6) Bonus. Find the charge density induced on either the cylinder or on the plane, as
a function of angular or linear coordinate on each, respectively. (Do only one or the
other.)
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Useful Formulas
Legendre Polynomials
1 1 1
P0 (x) = 1, P1 (x) = x, P2 (x) = (3x2 − 1), P3 (x) = (5x3 − 3x), P4 (x) = (35x4 − 30x2 + 3).
2 2 8
Spherical Harmonics
r r r r
1 3 3 ±iϕ 5
Y0 0 = , Y1 0 = cos θ, Y1 ±1 = ∓ e sin θ, Y2 0 = (3 cos2 θ − 1).
4π 4π 8π 16π
r r s
15 ±iϕ 15 ±2iϕ 2 2l + 1
Y2 ±1 = ∓ e cos θ sin θ, Y2 ±2 = e sin θ, Yl 0 = Pl (cos θ).
8π 32π 4π
Expansions
∞ l ∞ X l l
1 r< 1 1 r<
Y ∗ (θ0 , φ0 )Yl m (θ, φ)
X X
= P (cos γ),
l+1 l
= 4π l+1 l m
|~r − ~r 0 | l=0 r> |~r − ~r 0 | l=0 m=−l
2l + 1 r >
