Questions and Answers of Electrodynamics

Work out the remaining five parts to Eq. 12.117.
Prove that the symmetry (or antisymmetry) of a tensor is preserved by Lorentz transformation (that is: if t μv is symmetric, show that tμv is also symmetric, and likewise for antisymmetric).
Recall that a covariant 4-vector is obtained from a contravariant one by changing the sign of the zeroth component. The same goes for tensors: When you "lower an index" to make it covariant, you
A straight wire along the z axis carries a charge density λ traveling in the + z direction at speed v. Construct the field tensor and the dual tensor at the point (x, 0, 0).
Obtain the continuity equation (12.125) directly from Maxwell's equations (12.126).
Show that the second equation in (12.126) can be expressed in terms of the field tensor Fμv as follows:
Work out, and interpret physically, the μ = 0 component of the electromagnetic force law, Eq. 12.127.
You may have noticed that the four-dimensional gradient operator ∂/∂xμ functions like a covariant 4-vector--in fact, it is often written ∂μ, for short. For instance, the continuity equation,
Show that the potential representation (Eq. 12.132) automatically satisfies ∂Gμv / ∂xv = 0.
Inertial system S moves at constant velocity v = βc (cos Ф x + sin Ф y) with respect to S. Their axes are parallel to one another, and their origins coincide at t = t = 0, as usual. Find the
Calculate the threshold (minimum) momentum the pion must have in order for the process π + p → K + Σ to occur. The proton p is initially at rest. Use mπc2 = 150, mKc2 = 500, mpc2 = 900, mΣc2 =
A particle of mass m collides elastically with an identical particle at rest. Classically, the outgoing trajectories always make an angle of 90o. Calculate this angle relativistically, in terms of
Find x as a function of t for motion starting from rest at the origin under the influence of a constant Minkowski force in the x direction. Leave your answer in implicit form (t as a function of x).
An electric dipole consists of two point charges (±q), each of mass m, fixed to the ends of a (mass less) rod of length d. (Do not assume d is small.)(a) Find the net self-force on the dipole when
An ideal magnetic dipole moment m is located at the origin of an inertial system ,q that moves with speed v in the x direction with respect to inertial system S. In S the vector potential is (Eq.
An ideal magnetic dipole moment m is located at the origin of an inertial system ,q that moves with speed v in the x direction with respect to inertial system S. In S the vector potential is (Eq.
In a certain inertial frame S, the electric field E and the magnetic field B are neither parallel nor perpendicular, at a particular space-time point. Show that in a different inertial system S
Two charges +q approach the origin at constant velocity from opposite directions along the x axis. They collide and stick together, forming a neutral particle at rest. Sketch the electric field
"Derive" the Lorentz force law, as follows: Let charge q be at rest in S, so F = qE, and let S move with velocity v = vx with respect to S. Use the transformation rules (Eqs. 12.68 and 12.108) to
A charge q is released from rest at the origin, in the presence of a uniform electric field E = E0z and a uniform magnetic field B = B0x. Determine the trajectory of the particle by transforming to a
(a) Construct a tensor Dμv (analogous to Fμv), out of D and H. Use it to express Maxwell's equations inside matter in terms of the free current density Jμf.(b) Construct the dual tensor Hμv
Use the Larmor formula (Eq. 11.70) and special relativity to derive the Lienard formula (Eq. 11.73).
The natural relativistic generalization of the Abraham-Lorentz formula (Eq. 11.80) would seem to be This is certainly a 4-vector, and it reduces to the Abraham-Lorentz formula in the non-relativistic
Generalize the laws of relativistic electrodynamics (Eqs. 12.126 and 12.127) to include magnetic charge.
From Figure, approximate the(a) Smaller and(b) Larger wavelength at which the eye of a standard observer has half the eye's maximum sensitivity. What are the(c) Wavelength,(d) Frequency, and(e)
About how far apart must you hold your hands for them to be separated by 1.0 nano-light-second (the distance light travels in 1.0 ns)?
A certain helium-neon laser emits red light in a narrow band of wavelengths centered at 632.8 nm and with a "wavelength width" (such as on the scale of Figure) of 0.0100 nm. What is the corresponding
Project Seafarer was an ambitious program to construct an enormous antenna, buried underground on a site about 10 000 km2 in area. Its purpose was to transmit signals to submarines while they were
What inductance must be connected to a 17 pF capacitor in an oscillator capable of generating 550 nm (i e, visible) electromagnetic waves? Comment on your answer.
What is the wavelength of the electromagnetic wave emitted by the oscillator-antenna system of Figure if L = 0.253μH and C = 25.0 pF?
Some neodymium-glass lasers can provide 100 TW of power in 1.0 ns pulses at a wavelength of 0.26μm. How much energy is contained in a single pulse?
A plane electromagnetic wave has a maximum electric field magnitude of 3.20 x 10-4 V/m. Find the magnetic field amplitude.
A plane electromagnetic wave traveling in the positive direction of an x axis in vacuum has components Ex = Ey = 0 and E z = (2.0 Vim) cos [(π x 1015 s-1) (t – x/c)].(a) What is the amplitude of
In a plane radio wave the maximum value of the electric field component is 5.00 V/m. Calculate. (a) The maximum value of the magnetic field component and(b) The wave intensity.
What is the intensity of a traveling plane electromagnetic wave If B m is 1.0 X 10-4 T?
Assume (unrealistically) that a TV station acts as a point source broadcasting isotropic ally at 1.0 MW. What is the intensity of the transmitted signal reaching Proxima Centauri, the star nearest
The maximum electric field 10 m from an isotropic point source of light is 2.0 V/m. What are?(a) The maximum value of the magnetic field and(b) The average intensity of the light there?(c) What is
Frank D. Drake, an investigator in the SETI (Search for Extra-Terrestrial Intelligence) program, once said that the large radio telescope in Arecibo, Puerto Rico (Figure), "can detect a signal which
Sunlight just outside Earth's atmosphere has an intensity of 1.40kW/m2. Calculate(a) E m and(b) B m for sunlight there, assuming it to be a plane wave.
An electromagnetic wave with frequency 4.00 x 1014 Hz travels through vacuum in the positive direction of an x axis. The wave has its electric field directed parallel to the y axis, with amplitude E
An airplane flying at a distance of 10 km from a radio transmitter receives a signal of intensity 10μW/m2. What is the amplitude of the (a) Electric and (b) Magnetic component of the signal at
An isotropic point source emits light at wavelength 500 nm, at the rate of 200 W. A light detector is positioned 400 m from the source. What is the maximum rate ð B/ð t at which the magnetic
What is the radiation pressure 1.5 m away from a 500 W lightbulb? Assume that the surface on which the pressure is exerted faces the bulb and is perfectly absorbing and that the bulb radiates
A black, totally absorbing piece of cardboard of area A = 2.0 cm2 intercepts light with an intensity of 10 W /m2 from a camera strobe light. What radiation pressure is produced on the cardboard by
High-power lasers are used to compress plasma (a gas of charged particles) by radiation pressure. A laser generating radiation pulses with peak power 1.5 x 103 MW is focused onto 1.0 mm2 of
Radiation from the Sun reaching Earth (just outside the atmosphere) has an intensity of 1.4kW/m2.(a) Assuming that Earth (and its atmosphere) behaves like a flat disk perpendicular to the Sun's rays
Prove, for a plane electromagnetic wave that is normally incident on a flat surface, that the radiation pressure on the surface is equal to the energy density in the incident beam.(This relation
A small laser emits light at power 5.00mW and wavelength 633 nm. The laser beam is focused (narrowed) until its diameter matches the 1266 nm diameter of a sphere placed in its path. The sphere is
A plane electromagnetic wave, with wavelength 3.0 m, travels in vacuum in the positive direction of an x axis. The electric field, of amplitude 300 V/m, oscillates parallel to the y axis. What are
It has been proposed that a spaceship might be propelled in the solar system by radiation pressure, using a large sail made of foil. How large must the surface area of the sail be if the radiation
A small spaceship whose mass is 1.5 x 103 kg (including an astronaut) is drifting in outer space with negligible gravitational forces acting on it. If the astronaut turns on a 10 kW laser beam, what
In Figure, a laser beam of power 4.60 W and diameter D = 2.60 mm is directed upward at one circular face (of diameter d
Someone plans to float a small, totally absorbing sphere 0.500 m above an isotropic point source of light, so that the upward radiation force from the light matches the downward gravitational force
The intensity I of light from an isotropic point source is determined as a function of distance r from the source. Figure gives intensity I versus the inverse square r-2 of that distance. The
As a comet swings around the Sun, ice on the comet's surface vaporizes, releasing trapped dust particles and ions. The ions, because they are electrically charged, are forced by the electrically
In Figure, a beam of unpolarized light, with intensity 43 W/m2, is sent into a system of two polarizing sheets with polarizing directions at angles θ1 =70? and θ2 = 90? to the y axis. What is the
In Figure, a beam of light, with intensity 43 W/m2 and polarization parallel to a y axis, is sent into a system of two polarizing sheets with polarizing directions at angles of ?1 = 70? and ?2 = 90?
In Figure, initially unpolarized light is sent into a system of three polarizing sheets whose polarizing directions make angles of θ1 = θ2 = θ3 = 50? with the direction of the y axis. What
In Figure, initially unpolarized light is sent into a system of three polarizing sheets whose polarizing directions make angles of ?1 = 40?, ?2 = 20?, and ?3 = 40? with the direction of the y axis.
In Figure, unpolarized light is sent into a system of two polarizing sheets. The angles θ1 and θ2 of the polarizing directions of the sheets are measured counter clockwise from the positive
Unpolarized light of intensity 10mW/m2 is sent into a polarizing sheet as in Figure. What are?(a) The amplitude of the electric field component of the transmitted light and(b) The radiation pressure
At a beach the light is generally partially polarized due to reflections off sand and water. At a particular beach on a particular day near sundown, the horizontal component of the electric field
A beam of polarized light is sent into a system of two polarizing sheets. Relative to the polarization direction of that incident light, the polarizing directions of the sheets are at angles θ for
In Figure, unpolarized light is sent into a system of three polarizing sheets. The angles ?1, ?2 and ?3 of the polarizing directions are measured counterclockwise from the positive direction of the y
A beam of partially polarized light can be considered to be a mixture of polarized and unpolarized light. Suppose we send such a beam through polarizing filter and then rotate the filter through
In Figure, unpolarized light is sent into a system of three polarizing sheets, which transmits 0.0500 of the initial light intensity. The polarizing directions of the first and third sheets are at
We want to rotate the direction of polarization of a beam of polarized light through 90° by sending the beam through one or more polarizing sheets.(a) What is the minimum number of sheets
In Figure, unpolarized light is sent into a system of three polarizing sheets. The angles θ1 θ2 and θ3 of the polarizing directions are measured counterclockwise from the positive direction of the
Light in vacuum is incident on the surface of a glass slab. In the vacuum the beam makes an angle of 32.0° with the normal to the surface, while in the glass it makes an angle of 21.0° with the
In Figure a, a light ray in water is incident at angle ?1 on a boundary with an underlying material, into which some of the light refracts. There are two choices of underlying material. For each, the
Figure shows light reflecting from two perpendicular reflecting surfaces A and B. Find the angle between the incoming ray i and the outgoing ray r'.
In Figure a, a light ray in an underlying material is incident at angle θ1 on a boundary with water, and some of the light refracts into the water. There are two choices of underlying material. For
When rectangular metal tank in Figure is filled to unknown liquid, observer O, with eyes level the tank, can just see corner E. A ray that O at the top surface of the liquid is shown if D = 85.0cm
In Figure a, a beam of light in material 1 is incident on a boundary at an angle of θ1 = 30?. The extent of refraction of the light into material 2 depends, in part, on the index of refraction n2 of
In Figure, a 2.00-m-long vertical pole extends from the bottom of a swimming pool to a point 50.0 cm above the water. Sunlight is incident at angle ? = 55.0?. What is the length of the shadow of the
Dispersion in a window pane in Figure a beam of white light is incident at angle θ = 50? on a common window pane (shown in cross section) For the pane's type of glass, the index of refraction for
In Figure, light is incident at angle θ1 = 40.1? on a boundary between two transparent materials. Some of the light travels down through the next three layers of transparent materials while some of
In Figure a, a beam of light in material 1 is incident on a boundary at an angle θ1 = 40?. Some of the light travels through materials, and then some of it emerge into material 3. The two boundaries
In Figure a ray is incident on one face of a triangular glass prism in air. The angle of incidence θ is chosen so that the emerging ray also makes the same angle θ with the normal to the other
Rainbows from square drops suppose that, on some surreal world, raindrops had a square cross section and always fell with one face horizontal. Figure shows such a falling drop, with a white beam of
A point source of light is 80.0 cm below the surface of a body of water, find the diameter of the circle at the surface through which light emerges from the water.
The index of refraction of benzene is 1.8. What is the critical angle for a light ray traveling in benzene toward a flat layer of air above the benzene?
In Figure, light initially in material 1 refracts into material 2, crosses that material, and is then incident at the critical angle on the interface between materials 2 and 3. The indexes of
A catfish is 2.00 m below the surface of a smooth lake.(i). What is the diameter of the circle on the surface through which the fish can see the world outside the water?(ii). If the fish descends,
In the ray diagram of Figure, where the angles are not drawn to scale, the ray is incident at the critical angle on the interface between materials 2 and 3. Angle Φ = 60.0?, and two of the indexes
In Figure, light from ray A refracts from material 1 (n1 = 1.60) into a thin layer of material 2 (n2 = 1.80), crosses that layer, and is then incident at the critical angle on the interface between
Figure depicts a simplistic optical fiber: a plastic core (n1 = 1.58) is surrounded by a plastic sheath (n2 = 1.53). A light ray is incident on one end of the fiber at angle ?. The ray is to undergo
In Figure, a light ray in air is incident at angle ?1 on a block of transparent plastic with an index of refraction of 1.56. The dimensions indicated are H = 2.00 cm and W = 3.00 cm. The light passes
A catfish is 2.00 m below the surface of a smooth lake.(a) What is the diameter of the circle on the surface through which the fish can see the world outside the water?(b) If the fish descends, does
Suppose the prism of Figure has apex angle ? = 60.0? and index of refraction n = 1.60.(a) What is the smallest angle of incidence ? for which a ray can enter the left face of the prism and exit the
In Figure, light enters a 90? triangular prism at point P with incident angle θ, and then some of it refracts at point Q with an angle of refraction of 90?. (a) What is the index of refraction of
(a) At what angle of incidence will the light reflected from water be completely polarized?(b) Does this angle depend on the wavelength of the light?
Light traveling in water of refractive index 1.33 is incident on a plate of glass with index of refraction 1.53. At what angle of incidence is the reflected light fully polarized?
In Figure, a light ray in air is incident on a flat layer of material 2 that has an index of refraction n2 = 1.5. Beneath material 2 is material 3 with an index of refraction n3. The ray is incident
Rainbow figure shows a light ray entering and then leaving a falling, spherical raindrop after one internal reflection (see Figure a). The final direction of travel is deviated (turned) from the
The primary rainbow described in Problem 71 is the type commonly seen in regions where rainbows appear. It is produced by light reflecting once inside the drops. Rarer is the secondary rainbow
In Figure, a light ray enters a glass slab at point A at incident angle ?1 = 45.0? and then undergoes total internal reflection at point B. What minimum value for the index of refraction of the glass
In Figure, unpolarized light with an intensity of 25 W/m2 is sent into a system of four polarizing sheets with polarizing directions at angles θ1 = 40?, θ2 = 20, θ3 = 20?, and θ4 = 30?. What is
(a) Prove that a ray of light sheet of plate glass of thickness t face parallel to its initial direction but displaced sideways, as in Figure (b) Show that, for small angles of incidence θ, this
In Figure, two light rays pass from air through five layers of transparent plastic and then back into air. The layers have parallel interfaces and unknown thicknesses; their indexes of refraction are

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