A long, straight, solid cylinder, oriented with its axis in the z-direction,
carries a current whose current density is j. The current density, although
symmetrical about the cylinder axis, is not constant and varies according to the
relationship
j = (b/r) e(r-a)/δ
k for r < a, j
= 0 for r > a,
where the radius of the cylinder is a = 5.00 cm, r is the radial
distance from the cylinder axis, b is a constant equal to 600 A/m, and δ is
a constant equal to 2.50 cm.
(a) What is the total current I0 passing through the entire
cross section of the wire?
(b) Using Ampere's law, derive an expression for the magnetic field
B in the region r ≥ a.
Express your answer in terms of I0
(c) Obtain an expression for the current I contained
in a circular cross section of radius r ≤ a and
centered at the cylinder axis. Express your answer in terms of I0,
rather than b.
(d) Using Ampere's law, derive an expression for the magnetic field
B in
the region r ≤ a. Express your answer in terms of I0,
rather than b.
(e) Evaluate the magnitude of the magnetic field at r = δ,
a, and 2a.
A copper
rod slides on frictionless rails in the presence of a constant magnetic field B
= B0k. At t = 0 the rod is moving in the
y-direction with velocity v0.
(a) What is the subsequent velocity of the rod, if σ is its conductivity
and ρm is the mass density of copper? (Assume that the
resistance of the rails can be neglected.)
(b) For copper σ = 59.6*106 (Ωm)-1
and ρm = 8.9 g/cm3. If B0
is 1 gauss, estimate the time it takes the rod to stop.
(c) Show that the rate of decrease of the kinetic energy of the rod per
unit volume is equal to the ohmic heating rate per unit volume.
A magnetic dipole p is located at the origin. It lies in the x-z plane
and makes an angle θ with the z axis. At t = 0 a charge q is located at x = -x0
and is moving with velocity v0 in the positive y-direction.
Find the force the dipole exerts on q at t = 0.
A particle of charge q and mass m
moves in a region containing a uniform electric field
E = Ei pointing in the x-direction and a uniform magnetic field
B = Bk pointing in the z-direction.
(a) Write down the equation of
motion for the particle and find the general solutions for the Cartesian
velocity components vi(t) in terms of B, E, q, and m.
Hint: Let ζ = vx + ivy, and solve for ζ(t).
(b) Solve for the position of the particle as a function of time if the
particle is released from rest at t = 0.
(a) Find the magnetic field at any point in the x-y
plane for y > 0 due to a wire of length l carrying a current I from x = 0
to x = l along the x-axis.
(b) Use the result of (a) to find the self inductance of a square current loop of
side l. You may leave your result as a definite integral, but the limits must be
specified and the integrand must be in terms of the dimensions of the loop, constants, and
the variables being integrated.
(c) A conducting square current loop of side l with sides parallel to the x- and y-axis
has resistance R and self inductance L. It moves at a speed v in the +x-direction. The loop passes through a magnetic field given by
B = B0 k, -l/2< x < l/2, B = 0 elsewhere. Find the current in the
wire as a function of time assuming that I = 0 a long time before the loop reaches
the magnetic field.