June 2004
ELECTRICITY AND MAGNETISM
PROBLEMS FROM THE
1. Below
a critical magnetic field Hc,
type I superconductors exhibit the Meissner effect, in which the magnetic
induction is excluded from the volume of the
superconductor. Assume that in this case
the superconductor in question has no holes – i.e., it is topologically like a
sphere and not like a ring.
(a) Explain why the component of normal to the surface vanishes just outside
the superconductor.
(b) Explain why the magnetic induction
outside the superconductor can be derived from a scalar potential, .
State the boundary value problem
(differential equation and boundary conditions) satisfied by .
(c) Consider a superconducting sphere of
radius a placed in an otherwise
uniform magnetic induction . Find the magnetic induction outside the
sphere.
2. Here
we deal with the classical theory of the width of an atomic spectral line. The
“atom” consists of an electron (mass m
and charge –e) in a harmonic oscillator
potential with a drag force acting on it, so that the classical equation of
motion is
.
In the first two parts, ignore any loss of energy due to radiation.
(a) Suppose that at time t=0,
x=x0, =0.
Assuming that the damping is small, i.e., that ,
what is the subsequent motion? (Hint: Try a solution of the form .)
(b) A classical electron executing the
motion in part (a) would emit electromagnetic radiation. Determine the
functional dependence on the frequency of the intensity distribution function .
Note that we are asking only for the -dependence of the answer, not for the full
normalization factors. [ is the total energy integrated over time
emitted in a range of frequencies from .] (Hint: Parseval’s theorem, ,
where is the fourier transform of x(t), may be useful here.)
(c) Now suppose that the damping force is completely absent from the equation of
motion and that the oscillation is damped only by the loss of energy due to
radiation. The energy of the oscillator decays as . Determine assuming that in one period the electron loses
only a small fraction of its energy.
(d) Using the above model, what is the width
of an “atomic” spectral line of 500 nm according to the result of part (c)? Useful data: .
3. The
electrostatic field energy U of a
capacitor can be expressed as a function of the charge Q on the electrodes (the conductors that carry the capacitor
charge) and the capacitance C; U may additionally depend on a parameter
x (e.g., the position of a dielectric
between the electrodes). If the
electrodes are isolated, so that Q is
fixed, then the generalized electrostatic force associated with a virtual
displacement if x is
(a) If instead of isolating the electrodes ,
the capacitor is attached to the poles of a battery of emf V, show that
(b) A long cylindrical capacitor is lowered
vertically (along its axis) into a pool of liquid dielectric of dielectric
constant and mass density .
If a voltage V is applied between the
inner cylinder (radius a) and the
outer cylinder (radius b) of the
capacitor, the liquid rises to a height h
above its initial value between them.
Show that
Potentially useful information:
4. An
electron (mass m, charge –e) is bound with and effective elastic
force .
A static uniform magnetic field is also present. The electron is sent into
forced non-relativistic motion by one of two circularly polarized waves whose
electric field is
,
where .
(a) Any transverse vector can be expressed
in a “circular-polarization basis” as .
Show that in this basis the electron’s steady-state position components are
given by
,
where .
(b) Circularly polarized electromagnetic
waves are propagating along the z-axis
in a medium composed of N electrons
per unit volume bond as above with a static uniform magnetic field present as
above. Show that the indices of refraction for the polarized waves satisfy
.
(c) After
propagating a distance l in a medium,
the polarization direction of a wave initially linearly polarized in the x-direction
rotates by an angle in the xy-plane. Describe qualitatively when this happens and
show that is given by
.
5. Consider a rectangular metallic waveguide, infinitely long
in the z-direction, and of width (x-direction) a and
height (y-direction) b. The lowest frequency wave that can
propagate corresponds to a “transverse electric” mode for which the only
non-zero electric field component is Ey.
(a) Assuming that the fields have (z,
t) dependence exp[i(kz - wt)], show that the transverse
fields for this mode are given in terms of Bz by
,
where .
(b) Show that Bz satisfies
the differential equation
.
Also, what are the boundary conditions satisfied by Bz?
(c) It is sometimes useful to be able to
insert a probe to sample the field at various z-positions along a
waveguide. By considering the wall currents for this mode, where should you cut
a slot aligned along the z-direction into which you could insert a
movable probe without substantially disturbing the current distribution and
thereby the fields? Explain your answer.
6. Two identical classical solid spheres, radius R, each
carrying a total charge q distributed uniformly through them, spin with
an angular frequency w. They are separated by
a distance r >> R. Their spin axes are parallel and aligned
with the z-axis, while the line between the spheres is along the x-axis.
(a) What is the dipole magnetic moment of either sphere?
(b) What is the (dipole) magnetic field at
the position of one sphere due to the second sphere? You can express your
answer in terms of ,
the separation of the spheres, and any other relevant variables.
(c) What is the force on one sphere due to
the other? You can express your answer in terms of ,
the separation of the spheres, and any other relevant variables.
(d) What is the electromagnetic interaction
energy of the two spheres? You can express your answer in terms of ,
the separation of the spheres, and any other relevant variables.
7. A rectangular region of space is specified by 0 < x
< a, 0 < y < b and 0 < z < c. Boundary conditions are imposed such that
1. the face at z = c is at potential V1,
2. the face at x = a is at potential V2 and
3. all other boundary faces are at zero potential .
Find the potential
everywhere inside the volume.
8. A simple model of the ionosphere is a low density gas
of N electrons (and positive
ions) per unit volume.
(a) Show that the ionosphere has an index of
refraction given by
where
(b) Consider the earth having such a medium
beginning abruptly at a height h = 300km and extending to infinity. A radio amateur operating at a wavelength of
21m finds that she can receive distant (i.e. beyond line-of-sight)
stations located more than 1000km away, but none closer. Calculate N the electron density. For simplicity you may assume a flat
earth. (Useful data ε0 = 8.9
x 10-12F/m, me = 9.1 x 10-31 kg.)
9. Circa 1898, Lienard derived an expression for the power
radiated by a charge in relativistic motion:
(a) What is the non relativisitic limit of
Lienard’s expression for the power radiated?
(b) Suppose that the charge is constrained
to move in a circular path and is moving relativistically. Derive an expression
for the radiated power of the particle in terms of the time rate of change of the relativistic
momentum, .
(c) Find the emitted power for a particle
whose acceleration is parallel to its direction of motion and compare your
result to the one you found for part (b).
(d) Consider a particle of mass m and
charge q moving in a plane perpendicular to a uniform static magnetic
field B. Give the expression for the total energy radiated per unit time
in terms of m, q, B and the ratio of the particle’s total
energy to its rest energy.
10. A charged particle with a speed v is traveling in a
medium of permittivity e, possibly dependent on
frequency; the magnetic properties of the medium are those of the vacuum.
(a) At what threshold speed will the
particle emit Cerenkov light?
(b) Derive an expression for the angle of
emission of the Cerenkov radiation in terms of the speed of the particle and
the permittivity. If you use a graphical method, show your diagram (carefully
labeled).
(c) Consider two media, one with index of
refraction n1 and a second with index of refraction n2,
where n1 > n2. In which medium will the
threshold velocity for emission of Cerenkov radiation be larger? Which medium
will have the larger Cerenkov angle at a given velocity?
(d) Is Cerenkov radiation coherent or
incoherent? Justify your answer.
(e) What is the state of polarization of
Cerenkov radiation? Argue for your answer. (Hint: Remember the electric
and magnetic fields generated by a charged particle moving with constant velocity past a stationary observer at
the position of the observer.)
(f) The energy loss per unit length of a
charged particle due to Cerenkov radiation is given by
.
What is the lower limit of the integral?
(g) Describe qualitatively in terms of the
number of photons and the total energy emitted per unit frequency interval, the
spectrum of the emitted Cerenkov photons. Assume the dispersion of the medium
is negligible.
(h) Give another example of a type of
radiation generated by a charged particle that does NOT require that the
particle be accelerated. Explain qualitatively how such radiation can be
produced and give an example of a medium that could generate such radiation.
11. An electromagnetic wave described by a vector potential
strikes a spinless charge e at rest. Here A(x - ct) is an arbitrary
function (not necessarily a plane wave) that vanishes as its argument |x - ct| ®¥. Find the components of
the velocity of the charge generated by the interaction with the wave. Treat
the problem non-relativistically but do not ignore the magnetic field.
12. A steady current I is flowing in a
rectangular loop of thin wire of negligible resistance as shown below:
(a) What
is the magnitude and direction of the magnetic moment generated by the loop?
(b) Consider
the current to be a stream of positive charges moving as shown in the diagram
above. If we impose an electric field in the plane of the loop, upward and parallel
to the short sides of the loop as shown, what will be the effect on the
velocities, accelerations and density of charges in the loop in each segment
AB, BC, CD, DA? (Assume the field is the same within and outside the wire.)
(c) What
will be the non-relativistic total momentum of the charges in the loop if each
charge carrier has mass M?
(d) What
will be the relativistic total momentum (magnitude and direction) of the charge
carriers in the loop if each charge carrier has rest mass M? Express
your answer in terms of .
(e) Is
there any other net linear momentum in the system beyond what you calculated
above? Explain your answer.
13. (a) Consider an infinitely long straight
cylindrical conductor of radius R and
magnetic permeability m,
with a constant I running along the cylinder and distributed uniformly across
it. Assuming that the surrounding
material is a vacuum, find the vector potential ,
the magnetic flux density ,
and the magnetic field everywhere.
(b) Two
cylinders like that of part (a), radii respectively, are parallel to one another with
their axes separated by a distance d. They carry equal and opposite currents. Find the self-inductance L per unit length of this system in the limit that the distance d is very large compared to the radii.
14. A
radiating particle is observed from two frames moving uniformly with respect to
one another. Compare the rate of energy
loss of the particle due to radiation in the two frames.
15. Consider a
point charge e that moves along the z-axis with speed v
and an observation (laboratory) point O at (x0, 0, 0). In the
following make sure you are consistent in your unit system.
(a) By
starting in the charge rest frame and transforming appropriately, find the
scalar and vector potentials f and in the laboratory frame.
(b) Find
the electric and magnetic fields in the laboratory.
(c) Sketch
the electric field in the lab frame, assuming v is a significant
fraction of c. As usual, your sketch should use higher field line
density as an indication of stronger fields.
16. Consider a
steady-state circuit driven by a battery of emf E in which the
overall resistivity is dominated by a particular resistor, and in which we need
consider no other circuit elements.
(a) Verify
conservation of energy in a volume that encloses only the resistor, not the
battery.
(b) Describe,
qualitatively, how your treatment of this problem would change if instead of a
steady state battery you had a harmonic generator E = V0coswt.
17. A
dielectric sphere with dielectric constant K of radius R has a free charge density r distributed uniformly throughout the volume.
(a) What
is the electrostatic potential at the center of the sphere, relative to
infinity?
(b) How
much energy is required to establish this configuration, starting with the
charge dispersed at infinity?
18. Derive the
expression for the skin depth d(w) in a good conductor with conductivity s and magnetic permeability m. What does
“good” conductor mean? Say the magnitude
of the electric field just inside a
surface of such a medium is E0. Is this electric field normal or tangential
to the surface? What is the power
dissipated per unit area of the surface?
19. A charge q is placed within a spherical cavity of
radius a in a conducting material.
r is the distance of the charge from
the center of the cavity. Points R and R¢
are arbitrary points outside the cavity.
(a) What
is the difference of potential between points R and R¢?
(b) Is the
force on q such as to attract it to
or repel it from the walls of the cavity? (Explain.)
(c) What
is the magnitude of the force?
20. The
A superconducting slab obeying the
outside the superconductor.
(a) What
is inside the superconductor as a function of z ?
(b) What
is inside the superconductor as a function of z?
21. A
localized distribution of charges with total charge e and quadrupole moment Q
is placed in a cylindrically symmetric electric field with gradient in units of
(a) Write
an expression for the energy of the quadrupole interaction W, in terms of e, Q, and .
(b) Given
the values: Q = 2 ´
10-24 cm2, W/h = 107 s-1
where h = 6.63 ´ 10-27 erg-s, and e
is the charge on the electron,
calculate in units of
22. The
conductance s is defined via J
= sE.
(a) Derive
that for a good conductor (s >> w), the dielectric constant is approximately (Gaussian
units)
(b) What
is the skin depth in such a case?
(c) A wave
of magnitude E0 and
frequency w is normally incident on such a conductor. What is the magnitude of B inside the conductor (as a function of distance from the edge)?
(d) What
is the total power (for a surface of area A)
being dissipated inside the conductor?
(You may use any standard units, as long as you get it right.)
23. An
insulating circular disk of radius a
has a static charge distribution of s which varies radially as kr2. The disk rotates about its center with an
angular velocity w. Find the magnetic field at the center of the
disk
24. (a) A particle moving transversely to the
line of sight at a relativistic velocity u
emits a photon of frequency n as seen in its rest frame. At what frequency is it detected?
(b) In a
colliding beam accelerator, two beams of particles with rest masses m1 and m2 and momenta p1
and p2 collide with an
angle q between them
(in the lab frame). What is the total
available energy in the frame with total momentum = 0, and what is the
direction and velocity of the center-of-mass in the lab frame?
25. Consider a
circular loop of radius a carrying current I, its axis being
coincident with the x axis and its center being at the origin. The plane of the loop is in the y-z plane.
(a) Find
the field at all points along the x axis.
(b) Using
the divergence of find an approximate formula for By
valid for small values of y.
(c) A
second loop coaxial with the first one, centered at x = L, and
carrying a current I′ in the opposite sense to I, has a radius b
sufficiently small so that the expression for in part (b) is valid. Find the net force exerted on the small loop
by the field of the first loop.
26. A particle
of mass m moves at relativistic speed
u in a circle of Radius R in the z = 0 plane, the
orbit being normal to a constant magnetic field B = in the z-direction.
(a) Find R in terms of the other parameters,
neglecting radiation.
(b) Consider
an observer moving along the y axis
at velocity u. At what point in the particle's orbit does
the observer see the particle instantaneously at rest?
(c) What
is the force on the particle at this point according to the observer?
27. Two long
coaxial cylindrical shells of radii a
and b (a < b) and each of
length h are held at a fixed potential difference .
Find the surface charge densities on the two cylinders in two cases (ignore end
effects and assume that the field outside the system is zero):
(a) The
space between the cylinders is empty.
(b) The
space between the cylinders has a volume charge density ,
where k is a constant.
28. An
oscillating electric dipole of moment is aligned along the z-axis.
(a) What
are the limiting forms of and in the radiation zone?
(b) Calculate
the time averaged power radiated per unit solid angle.
(c) Sketch
the angular distribution of or in X-Z plane.
(d) If the
frequency becomes so high that the charges are moving relativistically indicate
roughly, what will happen to the angular distribution of radiation.
29. A
spherical shell of radius R and constant surface charge density s is rotated with angular velocity w about a diameter.
The magnetic scalar potentials inside and outside the sphere will be given by
spherical harmonics of the form
,
Find the magnetic intensities, inside and, ,
outside the sphere.
30. A plane
electromagnetic wave propagating in the +z
direction is incident on a plane mirror made of perfectly conducting material
in the x-y plane.
(a) Express
the reflected magnetic and electric fields, in terms of the incident ones, .
(b) In the
half-space z<0, where are the
zeros of the magnetic and electric fields?
(c) What
is the Poynting vector in the half-space z<0
?
If the perfect conductor is now set in motion in the +z direction with a speed comparable to c,
(d) Express
the reflected magnetic and electric fields in terms of the incident ones, in
the lab system.
(e) Calculate
the reflected electric field in the limit;
31. Consider a
plane electromagnetic wave of amplitude and frequency w traveling along z and incident on a free charge e
located at the origin. E is oriented in
the x-direction.
(a) What
is the equation of motion of the charge?
(b) What
is the solution for the position of the charge in the nonrelativistic limit,
v/c << 1? Neglect effects of for the first order calculation.
(c) Calculate
the actual position of the charge which is the real part of the expression
obtained in part (b).
(d) Calculate
the dipole moment associated with the oscillating charge.
(e) Calculate
the emitted power per unit solid angle emitted from the dipole.
(f) What
is the value of the differential scattering cross-section defined as
(Problem of Thomson scattering)
32. Consider
an electron incident on a proton target with the threshold energy required to
produce a neutral
(a) Calculate
the threshold kinetic energy of the electron.
(b) Calculate
the velocity of the center-of-mass.
(c) If the
h decays into two g rays with one g
ray being emitted along the z-axis (directly forward), at what angle in the
laboratory will the second g ray be observed?
d) What
are the energies of the g
rays in the laboratory?
33. A
relativistic charge of value +q passes along the positive z-direction with
velocity at a distance b above the z-axis. A charge
of value +2q passes in the negative z direction with velocity v/2 at a distance
b below the z-axis.
(a) Calculate
the maximum electric and magnetic fields seen by an observer in the laboratory system at the
origin. Which particle produces the
dominant fields for
(b) Calculate
the fields seen by the observer as a function of time. Which particle produces the dominant fields as seen by the observer at long times?
(c) Show
in diagrams how the observed fields vary with time.
34. Think of a
transmission line (two long parallel perfect conductors of arbitrary, but
constant cross-section) as a lumped-constant circuit with inductance L per unit
length and capacitance C per unit length.
(a) Show
that the speed of propagation of long-wave disturbances is .
(b) Find L
and C for two long parallel strips of width b, separated by a distance a
<< b, in vacuo. Show that LC = ,
where c is the speed of light. Is this
result true for any geometry of the transmission line, or when is it true?
35. A
conducting disc of thickness h and radius a
(a >> h) is placed at the center of a loop of radius R, as shown.
An alternating current is applied to the loop. Taking the resistivity of the conducting disc
to be r, calculate the average rate of power loss, assuming a << R and that the loop is a perfect conductor.
36. Two point
charges are constrained to move along the x ,
axis with constant (but relativistic) velocities .
(a) Calculate the force on each charge at the
particular time when the two charges are a distance d apart.
(b) Determine
these forces for the following special cases:
(i)
(ii)
Comment on the applicability of
37. A
localized charge distribution consists of two solid, uniform hemispheres of
charge +Q and -Q and radii R.
With respect to the polar axis shown above:
(a) Identify the non-vanishing multipole moments of this distribution having the two
lowest l values.
(b) Calculate explicitly the leading term in
the expansion for the electrostatic potential for r > R, and display the dependence of the next non-vanishing term
(ignoring the constant coefficient).
(c) Repeat (a) and (b) for the same charge
distribution, but with respect to a new polar axis oriented perpendicular to the original axis.
State which spherical harmonics are involved.
Spherical harmonics
38. A
radiating system consists of a molecule with two charges ±q a distance a apart
rotating with angular velocity w around a perpendicular axis through the midpoint. For c/w >> a. what is the leading multipole radiation term? What is the pattern of the radiation field,
i.e., what is the angular dependence of the emitted power, summed over all
polarizations? (To leading order only).
39. (a) In the lab
frame, there are static fields at right angles to each other. Derive a parametric equation for the path of
a (relativistic) electron in this field configuration, assuming the electron's
initial velocity was perpendicular to the B field. Take neglect radiation.
(b) Give a brief qualitative description of
the electronic path, both for and
40. Bohr's correspondence principle states that in the limit of
large quantum numbers the classical power radiated in the fundamental is equal
to the product of the quantum energy and the reciprocal mean lifetime of the
transition from principal quantum number n to (n-1).
(a) Using nonrelativistic approximations,
show that in a hydrogen-like atom the transition probability (reciprocal mean
lifetime) for a transition from a circular orbit of principal quantum number n
to (n-1) is given classically by
(b) For hydrogen compare the
classical value from (a) with the correct quantum-mechanical results for the
transitions 2p ® Is (1.6 x sec), 4f
®
3d (7.3 x sec), 6h ® 5g (6.1 x sec).
41. Consider a cavity within a material of dielectric constant e. An electric field has
been set up within the medium by some external source such that far from the
cavity the field is ,
where E is a constant. Find the electric field in the cavity for the
case that:
(i) the cavity is the
limiting case of a very long thin cylinder along the z-axis;
(ii) the cavity is the
limiting case of a thin disc oriented perpendicular to the z-axis;
(iii) the cavity is a
sphere. For this case find the field everywhere.
42. A plane electromagnetic wave of frequency w is incident at right
angles on the plane surface of a medium having conductivity s(w), and the real part of
the dielectric constant e » 1 (at frequency w). What is the pressure exerted by the radiation
on the conductor? What is the
penetration depth?
43. A magnetic dipole of moment is situated at a distance z from the plane
surface of a medium of permeability m, and its orientation
forms an angle q with the normal to the plane.
(a) What is the torque acting on the dipole?
(b) What is the stable equilibrium
orientation of the dipole?
44. Conductor A has an unknown capacitance and carries a known charge . Conductor B has a known capacitance and is initially grounded. It is brought close to A but not allowed to
touch it. The ground contact is broken,
B is removed and its charge is found to be . Next, B is discharged, kept insulated, and
brought close to A, in the same position as before. Contact is made between A and B, and the
charge on B is now found to be . Determine from these data.
45. A beam of unpolarized light is incident on a molecule. Find the angular distribution ofscattered
light in these two cases:
(a) The molecule has an isotropic
polarizability a.
(b) The molecule has an axis of rotational
symmetry that is fixed in the z direction, a|| and a^, are the
polarizabilities parallel and perpendicular to this axis.
[Of course (a) is a particular case of (b), however an explicit answer to part (a)
is expected, even if (b) is not fully worked out].
46. Displaced
emission lines found in the spectrum of SS433 suggest that this source is
spewing out two narrow high-speed jets of matter in opposite directions, and
that it rotates as shown in figure below.
GEOMETRY OF THE ROTATING-JET MODEL of SS 433 is laid out in this schematic diagram. The rotation axis of the object is inclined to the line of sight to the solar system by an angle of about 80 degrees. The jets themselves are in turn inclined by about 20 degrees to the rotation axis. The period of the rotation is 164 days. The velocity of the jets works out in this model to be about 78,000 kilometers per second. The component of this velocity projected along the line of sight varies periodically as the axis of the jets rotates; hence the Doppler-shifted emission lines in the spectrum of the object fluctuate between maximum and minimum.
(a) Show that the component along the line
of sight of the velocity of the upper jet is (it's up to you to identify v, a, f, and W with
the data in the figure). What is the
corresponding formula for the lower jet?
(b) Derive a formula for the Doppler shift, is the undisplaced wavelength.
(c) For each jet, sketch as a function of time, t. When is the same for the two jets. Then what is its value and what is that value
due to?
47. A circular current loop of area A rotates with angular
frequency w
around its diameter. The current, I, is
kept constant. Find the radiated power,
to leading order in .
(For a quick answer, exploit the equivalence between a small current loop and a
magnetic dipole.)
48. Consider a spherical electrical charge (or electric monopole) e, and a magnetic monopole g a distance a from each other. They are
at relative rest.
(a) Write down the electric and magnetic
fields. Make a sketch.
(b) Write
down the momentum and angular momentum densities of the electromagnetic field.
(c) Show
that there is an angular momentum L
of magnitude eg/c (in Gaussian units)
associated with the system. What is the
direction of L? For this part, it is enough to show that L is given by eg/c times a convergent integral independent of a, which turns out to be equal to one.
49. (a) Given the energy density and Poynting flux of the electromagnetic field in free space,
find a combination of and which is Lorentz invariant. Put c = 1.
(b) What
is the Lorentz transformation which makes and
parallel ( ¢ x ¢ = 0)? Is there an exception?
Hint: Try ,
a = arbitrary constant.
50. The fields
of a moving point charge e are:
(a) Derive
Larmor’s (nonrelativistic) formula for the total radiated power in terms of the
particle’s charge and acceleration.
(b) Write
down the relativistic generalization of Larmor’s formula. Comment on the question of relativistic
invariance and check that the nonrelativistic limit comes out correctly.
(c) Use (b)
to find the power radiated by a relativistic electron moving in a circular path
at constant speed.
(d) Find
the energy loss/revolution for an electron in a synchrotron with path radius
100m. Assume the energy E = 10GeV
is effectively constant during one revolution.
(Take mc2
= 0.5MeV, e2/(mc2) = 3 x 10-13cm)
51. (a) Determine the potential due to a
metallic sphere of radius R carrying a charge Q and a point
object carrying charge q situated a distance y from the center of
the sphere.
(b) What is the force between the sphere and the point
object?
(c) Suppose Q and q have the same sign.
What is the sign of the force? Explain all cases.
52. Two halves
of a long hollow conducting cylinder of inner radius b are separated by small lengthwise
gaps on each side, and are kept at different potentials and (see the accompanying figure). Find the potential inside in closed form.
53. Derive the
expression for the charge distribution giving rise to the following potentials:
(a) ,
where a and q are constants.
(b) ,
where and a are constants.
(c) Give a
physical interpretation of your answers.
What is the total charge in each case?
54. What is
the relativistic motion (trajectory) of a particle of charge e and mass m
moving in a uniform static electric field oriented in the x-direction? Assume an
initial momentum in the yz-plane.
55. A circular
ring of radius a with a uniformly distributed charge q lies on the x-y plane.
(a) Find
the electrostatic potential on the axis of the ring.
(b) Find
the potential at any point in space, but for large distances (r > a) from
the ring, keeping the first two terms.
(c) Repeat
(b) but for the following configuration: two coaxial circular rings of radii a and b (a> b) with uniformly
distributed charges q and -q respectively, lying in a given plane.
(d) Compare
the result with the potential due to a linear quadrupole.
56. Two
infinite, grounded, conducting planes are located at x = a/2 and x = -a/2. A point charge q is placed between the planes
at the point (x',y',z'), where -(a/2) < x' < (a/2).
(a) Find
the location and magnitude of all the image charges needed to satisfy the
boundary
conditions on the potential.
(b) Write
down the Green's function G .
57. A thin
metal rod of mass M, length l and cross-section A is spinning at a frequency w around a symmetrical perpendicular axis. Imagine that there is a slight excess of
electrons pushed toward the ends by centrifugal force.
(a) Make a
rough estimate of the total electromagnetic radiated power.
(b) The
total gravitational radiated power is ~ . Compare the two cases when the rod has a
density r = 10 gm/cm2 and w = I kHz. In G
= c = 1 units, (m/e) ~ 0.5 x 10-21, r ~ 10-27, w ~ 1/3 x 10-7.
58. An
oscillating electric dipole of moment is aligned along the z-axis.
(a) What
are the limiting forms of in the radiation zone?
(b) Calculate
the time-average power radiated per unit solid angle.
(c) For
the center fed, linear antenna shown in the figure below, calculate first the
magnitude of its dipole moment if the charge density r(z) is equal to ± 2iIo/wd (+ for positive z, - for negative z) and then its
time-averaged power radiated per unit solid angle.
59. Three
identical disks of radius a and negligible thickness are stacked parallel to each other, the
vertical z axis coinciding with their
common axis of symmetry. The separation
between the two outer disks is 2b and they are equidistant from the center disk
which is located with its center at the origin of coordinates. Each outer disk bears a total charge Q uniformly distributed over its
surface, whereas the center disk bears a total charge -2Q, also uniformly distributed over its surface.
(a) Compute
the Cartesian components of the electric quadrupole tensor
Qij, i, j = 1,2,3,
corresponding to the above charge configuration.
(b) Write
down the corresponding contribution to the electrostatic potential.
60. (a) Derive the expressions for in the far zone due to an oscillatory electric
dipole.
(b) An
electric charge q is arranged in a continuous spherically symmetric
distribution in a bounded region and executes radial pulsations. Find outside the charge distribution.
61. A small
test particle (mass m, positive charge q) makes circular orbits of radius R
around a "fixed"
(i.e. very massive) body of positive charge Q. A uniform magnetic field perpendicular to the orbital plane serves to
keep the particle in orbit. In the
inertial frame in which the central body is at rest, the rest charge is seen to
circle in the plane perpendicular to the B field with an angular frequency w. Determine the charge
to mass ratio of the test particle in terms of w, R, B, Q.
62. Ohm's law is satisfied in the rest frame of a conducting medium.
Derive the covariant generalization of Ohm's law in terms of the
4-current density ,
the field strength tensor and the 4-velocity of the medium .
63. A small loop of area A1, and current I1,
is placed at the origin of coordinates, with its positive normal in the
z-direction. At the point (r,q) on the x-z plane,
place a second small loop of area A2 and current I2, with
its positive normal lying in the x-z plane and making an angle a with the z-axis.
(a) Compute the mutual potential energy U21
Of loop (2) in the presence of loop (1).
(b) Assume that loop (1) is rigidly
fixed. Compute the components Fr
and Fq
of the force acting on loop (2) when a= 0, i.e., when both
magnetic dipoles have parallel moments.
(c) Compute the torque acting on loop (2),
about an axis perpendicular to the x-z plane, when a = 0 and loop (1) is
rigidly fixed.
64. Neglecting "retardation" effects and making use of the integral form of
Ampere's law*, show that the magnitude of the magnetic field produced by a
point charge in uniform rectilinear motion is given by
,
where q is the charge of the particle, u is its (constant) speed, r is the
distance from the instantaneous position of the particle to the fixed point of
observation, and q is the angle that makes with the (forward) direction of motion.