Download PHYS4210 Electromagnetic Theory Spring 2009 Midterm Exam #2

Name:
PHYS4210
Electromagnetic Theory
Midterm Exam #2
Spring 2009
Thursday, 26 March 2009
This exam has four questions and you are to work all of them. You must hand in your
paper by the end of class time (3:50pm) unless prior arrangements have already been made
with the instructor.
Note that not all of the problems are worth the same number of points.
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 absolutely necessary.
Good luck!
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Total:
Problem 1 (30 points). An electric field in free space is given by
E(r, t) = E0 î cos(kz − ωt)
a. (10 points) Show that E(r, t) satisfies the wave equation, given the appropriate relation
between k and ω.
b. (10 points) Find the field B(r, t) implied by Maxwell’s equations.
c. (10 points) A plane surface with area A lies in the xy plane. How much energy is carried
through this plane in a time T ? (You can assume that T 2π/ω so that averaging over
time is valid.)
Problem 2 (20 points). A particle with charge q moves in the x−direction with velocity
v = v î. It enters a region with constant electric and magnetic fields E = E ĵ and B = B k̂.
a. (10 points) Find the speed v such that the charge moves undeflected through the region
where the fields are nonzero.
b. (10 points) Use the Lorentz transformation properties of the fields to find the speed v
such that an observer traveling with the charge observes no force on the charge.
Problem 3 (25 points). An infinite conducting sheet lies in the xy plane. It carries a
uniform surface current density K = K î, where K is measured in current per unit length in
the y−direction. Find the magnitude and direction of the magnetic field as a function of z,
for both z > 0 and z < 0. Hint: Draw a rectangular amperian loop of length ` with height z
above or below the xy plane.
Problem 4 (25 points). At a certain instant of time, a charge q sits at the origin, i.e.
r = 0, with zero velocity but acceleration a in the z− direction.
a. (10 points) Write down the electric field E(r) at this time, as the sum of two terms, each
with a different dependence on r = |r|. Identify which term corresponds to the static electric
field, and which corresponds to the radiated electric field.
b. (5 points) Write down the magnetic field B(r) at this time.
c. (10 points) Find the Poynting vector at this time, as a function of r and polar angle θ.