Questions and Answers of Electrodynamics

The geometry of a two-dimensional potential problem is defined in polar coordinates by the surfaces ? = 0, ? = ?, and ?? = a, as indicated in the sketch. Using separation of variables in polar
A point charge q is located at the point (??, ?', z') inside a grounded cylindrical box defined by the surfaces z = 0, z = L, ? = a. Show that the potential inside the box can be expressed in the
The walls of the conducting cylindrical box of Problem 3.23 are all at zero potential, except for a disc in the upper end, defined by ρ = b < a, at potential V.(a) Using the various forms of the
Consider the Green function appropriate for Neumann boundary conditions for the volume V between the concentric spherical surfaces defined by r = a and r = b, a Where gt(r, r') = rl/ rl+1>?+
Apply the Neumann Green function of problem 3.26 to the situation in which the normal electric field is Er = ?E0 cos ? at the outer surface (r = b) and is Er = 0 on the inner surface (r = a). (a)
A point dipole with dipole moment p is located at the point x0. From the properties of the derivative of a Dirac delta function, show that for calculation of the potential Ф or the energy of a
A nucleus with quadrupole moment Q finds itself in a cylindrically symmetric electric field with a gradient (?Ez/?z)0 along the z axis at the position of the nucleus. (a) Show that the energy of
A very long, right circular, cylindrical shell of dielectric constant ε/ε0 and inner and outer radii a and b, respectively, is placed in a previously uniform electric field E0 with its axis
A point charge q is located in free space a distance d from the center of a dielectric sphere of radius a (a < d) and dielectric constant ε /ε0.(a) Find the potential at all points in space as
Two concentric conducting spheres of inner and outer radii a and b, respectively, carry charges ?Q. The empty space between the spheres is half-filled by a hemispherical shell of dielectric (of
Two long, coaxial, cylindrical conducting surfaces of radii a and b are lowered vertically into a liquid dielectric. If the liquid raises an average height h between the electrodes when a potential
For each set of Stokes parameters given below deduce the amplitude of the electric field, up to an overall phase, in both linear polarization and circular polarization bases and makes an accurate
A plane wave is incident on a layered interface as shown in the figure. The indices of refraction of the three nonpermeable media are n1, n2, n3. The thickness of the intermediate layer is d. Each of
Two plane semi-infinite slabs of the same uniform, isotropic, nonpermeable, lossless dielectric with index of refraction n are parallel and separated by an air gap (n = 1) of width d. A plane
A plane-polarized electromagnetic wave of frequency w in free space is incident normally on the flat surface of a nonpermeable medium of conductivity a and dielectric constant ε.(a) Calculate the
A plane wave of frequency w is incident normally from vacuum on a semi-infinite slab of material with a complex index of refraction n(w) [n2(w) = ?(w)/?0]. (a) Show that the ratio of reflected power
The time dependence of electrical disturbances in good conductors is governed by the frequency-dependent conductivity G.58). Consider longitudinal electric fields in a conductor, using Ohm's law, the
A stylized model of the ionosphere is a medium described by the dielectric constant (7.59). Consider the earth with such a medium beginning suddenly at a height h and extending to infinity. For waves
Plane waves propagate in a homogeneous, non-permeable, but anisotropic dielectric. The dielectric is characterized by a tensor ?ij, but if coordinate axes are chosen as the principle axes, the
Use the Kramers-Kronig relation (7.120) to calculate the real part of ?(w), given the imaginary part of ?(w) for positive w as (a) Im ?/?0 = ?[?(w ? w1) - ?(w ? w2)], ? ? ? ? ? ?w2 > w1 >
Discuss the extension of the Kramers-Kronig relations (7.120) for a medium with a static electrical conductivity a. Show that the first equation in (7.120) is unchanged, but that the second is
A circularly polarized plane wave moving in the z direction has a finite extent in the x and ?? directions. Assuming that the amplitude modulation is slowly varying (the wave is many wavelengths
A transmission line consisting of two concentric circular cylinders of metal with conductivity a and skin depth ?, as shown, is filled with a uniform lossless dielectric (?, ?). ? ??? mode is
Transverse electric and magnetic waves are propagated along a hollow, right circular cylinder with inner radius R and conductivity a. (a) Find the cutoff frequencies of the various ТЕ and TM modes.
A waveguide is constructed so that the cross section of the guide forms a right triangle with sides of length a, a, ?2a, as shown. The medium inside has ?? = ?r = 1. (a) Assuming infinite
A resonant cavity of copper consists of a hollow, right circular cylinder of inner radius R and length L, with flat end faces.(a) Determine the resonant frequencies of the cavity for all types of
For the Schumann resonances of Section 8.9 calculate the Q values on the assumption that the earth has a conductivity σe and the ionosphere has a conductivity σi, with corresponding skin depths 8e
Apply the variational method of Problem 8.9 to estimate the resonant frequency of the lowest TM mode in a "breadbox" cavity with perfectly conducting walls, of length d in the z direction, radius R
A waveguide with lossless dielectric inside and perfectly conducting walls has a cross-sectional contour ? that departs slightly from a comparison contour Co whose fields are known. The difference in
To treat perturbations if there is a degeneracy of modes in guides or cavities under ideal conditions one must use degenerate-state perturbation theory. Consider the two-dimensional (waveguide)
(a) From the use of Green's theorem in two dimensions show that the TM and ТЕ modes in a waveguide defined by the boundary-value problems (8.34) and (8.36) are orthogonal in the sense that ∫A
The figure shows a cross-sectional view of an infinitely long rectangular waveguide with the center conductor of a coaxial line extending vertically a distance h into its interior at z = 0. The
An infinitely long rectangular waveguide has a coaxial line terminating in the short side of the guide with the thin central conductor forming a semicircular loop of radius R whose center is a height
A hollow metallic waveguide with a distortion in the form of a localized bend or increase in cross section can support nonpropagating ("bound state") configurations of fields in the vicinity of the
Using the Lienard-Wiechert fields, discuss the time-averaged power radiated per unit solid angle in nonrelativistic motion of a particle with charge e, moving(a) Along the z axis with instantaneous
A nonrelativistic particle of charge ze, mass m, and kinetic energy E makes a head-on collision with a fixed central force field of finite range. The interaction is repulsive and described by a
(a) Generalize the circumstances of the collision of Problem 14.5 to nonzero angular momentum (impact parameter) and show that the total energy radiated is given by where rmin is the closest
A nonrelativistic particle of charge ze, mass m, and initial speed v0 is incident on a fixed charge Ze at an impact parameter b that is large enough to ensure that the particle's deflection in the
A swiftly moving particle of charge ze and mass m passes a fixed point charge Ze in an approximately straight-line path at impact parameter b and nearly constant speed v. Show that the total energy
As in Problem 14.4a a charge e moves in simple harmonic motion along the z axis, z(t') = ? cos(w0t?). (a) Show that the instantaneous power radiated per unit solid angle is? where ? = ? w0/?. (b)
Show explicitly by use of the Poisson sum formula or other means that, if the motion of a radiating particle repeats itself with periodicity T, the continuous frequency spectrum becomes a discrete
(a) Show that for the simple harmonic motion of a charge discussed in Problem 14.12 the average power radiated per unit solid angle in the mth harmonic is (b) Show that in the nonrelativistic limit
A particle of charge e and mass m moves relativistically in a helical path in a uniform magnetic field B. The pitch angle of the helix is ? (? = 0 corresponds to circular motion). (a) By arguments
Consider the synchrotron radiation from the Crab nebula. Electrons with energies up to 1013eV move in a magnetic field of the order of 10?4 gauss. (a) For E = 1013eV, ? = 3 X 10?4 gauss, calculate
A radiating quadrupole consists of a square of side a with charges ± q at alternate corners. The square rotates with angular velocity ω about an axis normal to the plane of the square and through
Two halves of a spherical metallic shell of radius R and infinite conductivity are separated by a very small insulating gap. An alternating potential is applied between the two halves of the sphere
(a) Show that a classical oscillating electric dipole p with fields given by (9.18) radiates electromagnetic angular momentum to infinity at the rate (b) What is the ratio of angular momentum
(a) From the electric dipole fields with general time dependence of Problem 9.6, show that the total power and the total rate of radiation of angular momentum through a sphere at large radius r and
The transitional charge and current densities for the radiative transition from the m = 0, 2p state in hydrogen to the Is ground state are, in the notation of (9.1) and with the neglect of
An almost spherical surface defined byR(θ) = R0[1 + βP2(cos θ)]has inside of it a uniform volume distribution of charge totaling Q. The small parameter β varies harmonically in time at frequency
The uniform charge density of Problem 9.12 is replaced by a uniform density of intrinsic magnetization parallel to the z axis and having total magnetic moment M. With the same approximations as above
A thin linear antenna of length d is excited in such a way that the sinusoidal current makes a full wavelength of oscillation as shown in the figure. (a) Calculate exactly the power radiated per
Treat the linear antenna of Problem 9.16 by the multipole expansion method. (a) Calculate the multipole moments (electric dipole, magnetic dipole, and electric quadrupole) exactly and in the
A spherical hole of radius a in a conducting medium can serve as an electromagnetic resonant cavity. (a) Assuming infinite conductivity, determine the transcendental equations for the characteristic
If the light particle (electron) in the Coulomb scattering of Section 13.1 is treated classically, scattering through an angle ? is correlated uniquely to an incident trajectory of impact parameter b
Time-varying electromagnetic fields E(x, t) and B(x, t) of finite duration act on a charged particle of charge e and mass m bound harmonically to the origin with natural frequency ?0 and small
The external fields of Problem 13.2 are caused by a charge ze passing the origin in a straight-line path at speed v and impact parameter b. The fields are given by (11.152). (a) Evaluate the Fourier
(a) Taking h(ω) = 12Z eV in the quantum-mechanical energy-loss formula, calculate the rate of energy loss (in MeV/cm) in air at NTP, aluminum, copper, and lead for a proton and a mu meson, each with
Assuming that Plexiglas or Lucite has an index of retraction of 1.50 in the visible region, compute the angle of emission of visible Cherenkov radiation for electrons and protons as a function of
A magnetic monopole with magnetic charge g passes through matter and loses energy by collisions with electrons, just as does a particle with electric charge ze. (a) In the same approximation as
A nonrelativistic particle of charge e and mass m is bound by a linear, isotropic, restoring force with force constant mw20. Using (6.13) and (16.16) of Section 16.2, show that the energy and angular
A nonrelativistic electron of charge —e and mass m bound in an attractive Coulomb potential (–Ze2/r) moves in a circular orbit in the absence of radiation reaction.(a) Show that both the energy
(a) Show that the radiation reaction force in the Lorentz-Dirac equation of Problem 16.7 can be expressed alternatively as (b) The relativistic generalization of A6.10) can be obtained by replacing
(a) Show that for arbitrary initial polarization, the scattering cross section of a perfectly conducting sphere of radius a, summed over outgoing polarizations, is given in the long-wavelength limit
Electromagnetic radiation with elliptic polarization, described (in the notation of Section 7.2) by the polarization vector, is scattered by a perfectly conducting sphere of radius a. Generalize
A solid uniform sphere of radius R and conductivity σ acts as a scatterer of a plane-wave beam of unpolarized radiation of frequency ω, with ωR/c
Discuss the scattering of a plane wave of electromagnetic radiation by a nonper-meable, dielectric sphere of radius α and dielectric constant єr. (a) By finding the fields inside the sphere and
Consider the scattering of a plane wave by a nonpermeable sphere of radius ? and very good, but not perfect, conductivity following the spherical multipole field approach of Section 10.4. Assume that
The aperture or apertures in a perfectly conducting plane screen can be viewed as the location of effective sources that produce radiation (the diffracted fields). An aperture whose dimensions are
A perfectly conducting flat screen occupies half of the x-y plane (i.e., x 0 and wave number ? is incident along the z axis from the region z 0. Let the coordinates of the observation point be (X,
A linearly polarized plane wave of amplitude E0 and wave number k is incident on a circular opening of radius α in an otherwise perfectly conducting flat screen. The incident wave vector makes an
A rectangular opening with sides of length α and b > α defined by x = ±(α/2), у = ±(b/2) exists in a flat, perfectly conducting plane sheet filling the x-y plane. A plane wave is normally
(a) Show from (10.125) that the integral of the shadow scattering differential cross section, summed over outgoing polarizations, can be written in the short-wavelength limit as and therefore is
Discuss the diffraction due to a small, circular hole of radius a in a flat, perfectly conducting sheet, assuming that k? (a) If the fields near the screen on the incident side are normal E0e?i?t
Show explicitly that two successive Lorentz transformations in the same direction are equivalent to a single Lorentz transformation with a velocity v = v1 + v2/1 + (v1v2/c2)This is an alternative way
A possible clock is shown in the figure. It consists of a flashtube F and a photocell P shielded so that each views only the mirror M, located a distance d away, and mounted rigidly with respect to
A coordinate system K' moves with a velocity v relative to another system K. In K' a particle has a velocity u' and an acceleration a'. Find the Lorentz transformation law for accelerations, and show
Assume that a rocket ship leaves the earth in the year 2100. One of a set of twins born in 2080 remains on earth; the other rides in the rocket. The rocket ship is so constructed that it has an
In the reference frame К two very evenly matched sprinters are lined up a distance d apart on the у axis for a race parallel to the x axis. Two starters, one beside each man, will fire their
(a) Use the relativistic velocity addition law and the invariance of phase to discuss the Fizeau experiments on the velocity of propagation of light in moving liquids. Show that for liquid flow at a
An infinitesimal Lorentz transformation and its inverse can be written as where ??? and ??? are infinitesimal. (a) Show from the definition of the inverse that ??? = ? ??? . (b) Show from the
An infinitely long straight wire of negligible cross-sectional area is at rest and has a uniform linear charge density q0 in the inertial frame K'. The frame K' (and the wire) move with a velocity v
In a certain reference frame a static, uniform, electric field E0 is parallel to the x axis, and a static, uniform, magnetic induction B0 = 2E0 lies in the x-y plane, making an angle θ with the
In the rest frame of a conducting medium the current density satisfies Ohm's law J' = σE', where σ is the conductivity and primes denote quantities in the rest frame.(a) Taking into account the
The electric and magnetic fields of a particle of charge q moving in a straight line with speed v = ?c, given by (11.152), become more and more concentrated as ? ? 1, as is indicated in Fig. Choose
A particle of mass M and 4-momentum P decays into two particles of masses m1 and m2. (a) Use the conservation of energy and momentum in the form, p2 = P ? ?1, and the invariance of scalar products of
The presence in the universe of an apparently uniform "sea" of blackbody radiation at a temperature of roughly 3K gives one mechanism for an upper limit on the energies of photons that have
In a collision process a particle of mass m2, at rest in the laboratory, is struck by a particle of mass m1 momentum PLAB and total energy ELAB. In the collision the two initial particles are
In an elastic scattering process the incident particle imparts energy to the stationary target. The energy ?E lost by the incident particle appears as recoil kinetic energy of the target. In the
(a) A charge density ρ' of zero total charge, but with a dipole moment p, exists in reference frame K'. There is no current density in K'. The frame K' moves with a velocity v = βc in the frame K.
Show that the Lorentz invariant Lagrangian (in the sense of Section 12.1B)L = – mUαUα/2 – q/c UαAαgives the correct relativistic equations of motion for a particle of mass m and charge q
(a) Show from Hamilton's principle that Lagrangians that differ only by a total time derivative of some function of the coordinates and time are equivalent in the sense that they yield the same
A particle with mass m and charge e moves in a uniform, static, electric field E0. (a) Solve for the velocity and position of the particle as explicit functions of time, assuming that the initial
It is desired to make an E × В velocity selector with uniform, static, crossed, electric and magnetic fields over a length L. If the entrance and exit slit widths are ∆x, discuss the interval
A particle of mass m and charge e moves in the laboratory in crossed, static, uniform, electric and magnetic fields. E is parallel to the x axis; В is parallel to the у axis. (a) For | E | < | В |
The magnetic field of the earth can be represented approximately by a magnetic dipole of magnetic moment M = 8.1 ? 1025 gauss-cm3. Consider the motion of energetic electrons in the neighborhood of
Consider the precession of the spin of a muon, initially longitudinally polarized, as the muon moves in a circular orbit in a plane perpendicular to a uniform magnetic field B.(a) Show that the
An alternative Lagrangian density for the electromagnetic field is (a) Derive the Euler-Lagrange equations of motion. Are they the Maxwell equations? Under what assumptions?(b) Show explicitly, and
Consider the Proca equations for a localized steady-state distribution of current that has only a static magnetic moment. This model can be used to study the observable effects of a finite photon
(a) Starting with the Proca Lagrangian density (12.91) and following the same procedure as for the electromagnetic fields, show that the symmetric stress-energy-momentum tensor for the Proca fields

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