Midterm 1
Phy 182 - Fall 2010
Wednesday, Oct. 6, in class exam.
This exam contains 6 problems worth 20 points each. Your top 5 problems will be counted
towards your score. Useful equations are collected at the end of the exam.
Please sign a pledge on your exam stating that you have abided by Duke’s honor code.
Problem 1 (20 pts.)
Consider a sphere of radius R with uniform charge density within its volume such that
the total charge of the sphere is Q.
a) What is the electric field everywhere in space? Make sure you give the electric field
inside and outside the sphere and specify its direction.
b) Calculate the total electrostatic energy of this charge distribution.
Problem 2 (20 pts.)
Consider two concentric infinite conducting cylinders of radii a and b, a < b.
a) What is the capacitance per unit length of the cylinders?
b) What is the energy per unit length stored in the capacitor if the charge per unit length
on the inner cylinder is λ?
Problem 3 (20 pts.)
An ideal dipole with dipole moment p0 ẑ is placed at the center of a spherical cavity of
radius R within a large conductor.
a) What is the electrostatic potential within the cavity?
b) What is the induced charge on the surface of the spherical cavity?
Problem 4 (20 pts.)
A dielectric disc has radius R, thickness d, its axis of circular symmetry is aligned with
the z-axis, and its polarization is P0 ẑ. No other electric field or charges are present.
a) What is the bound surface charge σb and the bound volume charge ρb ? Specify where
these charges are located.
b) What is the exact electrostatic potential at a point on the z-axis a distance r above
the center of the disc?
c) Give an approximate expression for the electrostatic potential everywhere in space at
a distance r from the center of the disc when r ≫ R, d?
Problem 5 (20 pts.)
Solve the two-dimensional Laplace’s equation
!
∂2
∂2
+
V (x, y) = 0,
∂x2 ∂y 2
in the region 0 < x < a and −b < y < b with the boundary conditions
V (x, b) = −V (x, −b) = V0
V (0, y) = V (a, y) = 0
0 ≤ x ≤ a,
− b ≤ y ≤ b.
You can express your answer for the potential as an infinite series.
Problem 6 (20 pts.)
An ideal dipole with dipole moment p is a distance d above a conducting plane. The
dipole is oriented parallel to the surface of the conducting plane. What is the force on the
dipole? Give the magnitude and direction of the force.
p
d
Useful Formulae
Gauss’ Law
~ ·E
~ = ρ
∇
ǫ0
Z
~ = Qenc
d~a · E
ǫ0
Capacitors
1
W = CV 2
2
Q = CV
Solution to Laplace’s Equation in Spherical Coordinates
V (r, θ) =
Bl
Al r + l+1
r
∞ X
l=0
l
Pl (cos θ)
Multipole Expansion
1
V (~r ) =
4πǫ0
!
Q p~ · ~r
+ 3 + ...
r
r
Electric Field of an Ideal Dipole
~ dipole =
E
1 3(~p · r̂)r̂ − ~p
4πǫ0
r3
Vector Calculus (more on next page)
r̂ =
~r
r
~ = r̂
∇r
∇i ~rj = δij
~ · r̂ = 4πδ 3 (~r)
∇
r2
Fourier Transform
2
π
Z
0
π
dθ sin(n θ) sin(m θ) = δnm