Download PHYS4210 Electromagnetic Theory Spring 2009 Final Exam

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PHYS4210
Electromagnetic Theory
Final Exam
Spring 2009
Wednesday, 6 May 2009
This exam has two parts. Part I has 20 multiple choice questions, worth two points each.
Part II consists of six relatively short problems, worth ten points each. The short problems
can be worked on on the front page of the sheet provided, but use the back if you need more
room. In any case please be neat!
You may use your textbook, course notes, or any other reference you may have other
than another human. You are welcome to use your calculator or computer, although the test
is designed so that these are not necessary.
Good luck!
Part I (Total):
Part II
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Problem 5:
Problem 6:
Total:
Part I: Multiple Choice Questions (Two points each)
1. The electric field is evaluated at a certain point far from a localized charge distribution whose net charge is zero. At a point twice as far away, but in the same direction,
the electric field is
A. multiplied by 2.
B. multiplied by 1/2.
C. multiplied by 1/4.
D. multiplied by 1/8.
E. the same, since it must equal zero everywhere.
2. An infinitely large plane is charged by an amount σ per unit area. The electric
field magnitude at a distance r from the plane is
A. 0
B. σ
C. 2πσ
D. 4πσ
E. σ/r2
3. An electric field E(r, t) and a magnetic field B(r, t) are derived from the vector
and “scalar” potentials A(r, t) and φ(r, t). If I add the vector atî to A(r, t) (where a
is a constant and î is the unit vector in the x-direction), then what must I subtract
from φ(r, t) in order to keep E(r, t) and B(r, t) unchanged?
A. a
B. at
C. ax/c
D. axt/c
E. nothing.
4. Which of the following is equivalent to Coulomb’s Law?
A. ∇ · E = 4πρ
B. ∇ · B = 0
C. ∇ × E = −(1/c)∂B/∂t
D. ∇ × B = (1/c)∂E/∂t + 4πj
E. F = qE + (q/c)v × B
5. Given a vector function F(x, y, z) = xî + z ĵ + y k̂, and a box of side length a
sitting with one corner at (x, y, z) = (0, 0, 0) and in the octant where x, y, and z, are
all positive, the value of
I
F · da
box
that is, the surface integral over the box, is
A. 0
B. a2
C. a3
D. 6a2
E. 3a3
6. Five positive charges of magnitude q are arranged symmetrically around the circumference of a circle of radius r. What is the magnitude of the electric potential at
the center of the circle?
A. 0
B. q/r
C. 5q/r
D. q/r cos(2π/5)
E. 5q/r cos(2π/5)
7. Which of the following statements is false?
A. The electric field E is a polar vector in three-dimensional space.
B. The magnetic field B is an axial vector in three-dimensional space.
C. The inner product E · B of electric and magnetic fields is invariant under both
spatial rotations and Lorentz transformations.
D. The electric field E and magnetic field B specify the elements of a four-dimensional
tensor in space and time.
E. The electric field E is the same to all observers, but the magnetic field B depends
on the particular reference frame.
8. A wave component moving in the z-direction is written as the complex function
E(z, t) = E0 e−i(kz−ωt)
Which of the is the time-averaged value of the physical electric field?
A. 0
B. E0 /2
√
C. E0 / 2
D. E0
E. E02
9. As shown in class, the electric field from a static charge distribution ρ(r) is
E(r) =
Z
all space
ρ(r0 )(r − r0 ) 0
dV
|r − r0 |3
R
This implies that which of the following is true for a point charge Q = ρ(r)dV ?
I. ∇ · E = 4πρ
II. ∇ × E = 0
III. |E| = Q/|r|2
A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
10. A transverse, traveling electromagnetic plane wave with wavelength λ and (angular) frequency ω moves through a medium with (real) index of refraction n. Which
of the following equals the “speed of light” c?
A. λω
B. λω/2π
C. nλω
D. nλω/2π
E. λω/2πn
11. An oscillating magnetic dipole emits electromagnetic radiation. If r represents
the distance from the source, then the electric field at large distances falls like
√
A. 1/ r
B. 1/r
C. 1/r3/2
D. 1/r2
E. 1/r3
12. Two simultaneous events are separated in space by a distance L. In a different
reference frame, moving at a speed v relative to the√first, they are separated by ∆t in
time and ∆x in space. Writing β ≡ v/c and γ = 1/ 1 − β 2 , it is true that
A. L2 = (∆x)2
B. L2 = (γ∆x)2
C. L2 = (c∆t)2
D. L2 = (cγ∆t)2
E. L2 = (∆x)2 − (c∆t)2
13. A plane wave is incident on a plane boundary between two dielectric media, with
different indices of refraction. It is normally incident, that is, it moves perpendicularly
to the plane boundary. Which of the following has the same value on either side of the
boundary and very close to it?
I.
II.
III.
IV.
The
The
The
The
electric field parallel to the interface
electric field perpendicular to the interface
magnetic field parallel to the interface
magnetic field perpendicular to the interface
A. I only
B. III only
C. I and III only
D. II and IV only
E. I, II, III, and IV
14. Which of the following modes cannot be supported in a rectangular waveguide?
A. TE10
B. TE11
C. TE12
D. TM10
E. TM11
15. A reasonable approximation for the index of refraction of glass is given by
n=
v
u
u
t1 +
4πe2 N
m(ω02 − ω 2 )
where e and m are the electron charge and mass, and N is the electron density. If ω
corresponds to the angular frequency of visible light, then ω0 would be given by the
angular frequency of electromagnetic radiation
A. in the microwave region.
B. in the infrared region.
C. in the green region.
D. in the ultraviolet region.
E. in the X-ray region.
16. Two clean, uncharged parallel conducting plates are held close to each other in
a vacuum. They feel a force between them due to
A. rearrangement of atomic charges.
B. residual magnetic fields.
C. quantum mechanics.
D. thermodynamics.
E. nuclear forces.
17. A current I flows around a square loop with side length a. The magnetic dipole
has magnitude
A. Ia2 /2
B. Ia2
C. 2Ia2
D. 2πIa2
E. 4πIa2
18. A quantity Z is represented by a density η(r, t) and flows according to a velocity
field w(r, t). The mathematical statement that
∂η
+ ∇ · (ηw) = 0
∂t
is equivalent to stating that
A. Z = 0
B. Z is constant.
C. Z is conserved.
D. Z is independent of time.
E. Z is independent of space.
19. Three charges are arranged in a line. Two have charge +q and are separated by
a distance d. The third has charge −2q and is equidistant between the first two. This
arrangement has nonzero
A. total charge.
B. charge density.
C. dipole moment.
D. quadrupole moment.
E. dipole moment and quadrupole moment.
20. “Napolitano” is the last name of
A. the Secretary of Homeland Security.
B. the President of the Italian Republic.
C. the professor in charge of this course.
D. all of the above.
E. none of the above.
Part II: Short Problems (10 points each)
Problem 1. Show that, for spherical coordinate radius r, a function of the form
1
f (r) = e−i(kr−ωt)
r
satisfies the wave equation so long as v = ω/k is the speed of the wave.
Problem 2. A straight wire with circular cross section and radius R carries a nonuniform
current density J(r), where r is the distance from the axis of the wire. For J(r) = 3I0 r/4πR3 ,
find the magnetic field B(r) both inside (r < R) and outside (r > R) the wire.
Problem 3. Prove that if, in some reference frame, E is nonzero but B = 0, then there is
no reference frame in which E = 0. (Hint: Make use of a homework assignment.)
Problem 4. A spherical dielectric shell of inner radius a and outer radius b is uniformly
charged and carries a net charge Q. Find the electric potential as a function of the radius r,
for r < a, a < r < b, and r > b. You may leave the answer in terms of the charge density ρ,
but be sure to somewhere derive ρ in terms of Q, a, and b.
Problem 5. A “coaxial cable” is a waveguide made from two long coaxial conducting
cylinders. A “TEM” wave propagates with speed c along the cable (the z-direction) with
E(r, z, t) = r̂E(r)e−i(kz−ωt)
and
B(r, z, t) = θ̂B(r)e−i(kz−ωt)
using cylindrical coordinates r, θ, and z. Derive expressions for E(r) and B(r) and show
that they are consistent with Maxwell’s Equations.
Problem 6. A circular wire loop of radius R carries a current that oscillates sinusoidally
in time with an (angular) frequency ω. If the amplitude of oscillation is I0 , find the (time
averaged) power radiated per unit solid angle. Measure the solid angle relative to an axis
perpendicular to the plane of the current loop. You may refer to results derived in your
textbook, but clearly cite any equations that you use.