Problem 1:
A closed loop shown in the figure below, with a cross
section of 0.1 m2, is placed in an external magnetic field changing
with time as B = kt, where k is equal to 1 T/sec. The magnetic flux is
confined to the region inside the loop. R1 has resistance of 3
W and R2 has resistance of 2
W. Calculate
(a) the value of the current I in the circuit, and
(b) the readings of the voltmeters V1 and V2 and compare those values.
Problem 2:
(a) A sphere of radius R is filled uniformly with charge density
r. Sketch the electric field as a function of
distance r from the center of the sphere, for both r < R and r > R.
(b) Calculate the magnetic dipole moment m of a disk of radius R and
thickness d containing a uniform volume charge density r
and rotating with angular velocity w about its
symmetry axis.
(c) A square loop of wire of area A and resistance R is inside a constant
magnetic field B. Initially the loop is perpendicular to B.
It starts to rotate with angular velocity w about one of its sides. Find
an expression for the current flowing in the loop as a function of time.
(d) Derive the equation of continuity, ¶r/¶t
+ Ñ×j = 0, by applying charge conservation to
a small volume element.
Problem 3:
You need 12 V to run an electric train, but the outlet voltage is 120 V. What is the ratio of the number of turns on the primary coil to the number of turns on the secondary coil of the transformer you are using?
Problem 4:
A 3 m long solenoid with a diameter of 0.05 m has 3000 turns of wire with total
resistance of 1 W uniformly wound along its length. It
has an air core.
(a) Find the approximate self-inductance of this solenoid.
(b) If the solenoid and a 9 W resistor are
connected in parallel across the terminals of a 12 V battery, find the final
steady state current in the solenoid.
(c) If the battery is disconnected at time 0 leaving the circuit of the
solenoid and the resistor intact, find the current in the resistor as a function
of time.
(d) Show that the I2R losses in the resistor after
t = 0
is equal to the energy stored in the solenoid at t = 0.
Problem 5:
(a) Write down the set of Maxwell's equations in the form that applies
to static fields.
(b) Use the continuity equation to adapt Maxwell's equations to dynamic
fields. (This can be done by using a term that Maxwell referred to as
"displacement current".)
(c) Write down the dynamical form of Maxwell's equations.
(d) Introduce the vector and scalar potentials and show that this leads to
two wave equations in these potentials. (The Lorentz gauge may be used to
answer this part.)
(e) Define the Coulomb gauge and elaborate on some Coulomb gauge details.
Problem 6:
A toroidal coil of N turns has a square cross section, each side of
the square being of length a, and inner radius b.
(a) Find the self-inductance of the coil.
(b) Find the mutual inductance of the system consisting of the coil and
a long, straight wire along the axis of symmetry of the coil. (Assume that
the conductors closing the circuit of which the long straight wire is part of
are located far from the coil, so that their influence may be neglected.)
(c) Find the ratio of the self-inductance of the coil to the mutual
inductance of the system.
Problem 7:
Under the influence of the gravity near the surface of the
earth a square wire of length l, mass m and resistance R slides without friction
down very long parallel conducting rails of negligible resistance. The rails
are connected to each other at the bottom by a rail of negligible resistance,
parallel to the wire, so that the wire and the rails form a closed rectangular
conducting loop. The plane of the rails makes an angle
q with the horizontal, and a uniform
vertical magnetic field B exists throughout the region.
(a) Show that the wire acquires a steady state speed for any fixed
q.
(b) If the angle q is lowered from
q1 = p/3 to
q2 =
p/6, find the percent change of the steady-state speed of the
wire.
(c) Find the percent change of the corresponding power converted into joule
energy.
