Solved: A cylindrical resistor of length L, radius a, and resistance R carries current

Why is Maxwells modification of Ampres law essential to the existence of electromagnetic waves?

The presence of magnetic monopoles would require a modification of Gausss law for magnetism. Which other Maxwell equation would need modification?

Is there displacement current in an electromagnetic wave? Is there ordinary conduction current?

List some similarities and differences between electromagnetic waves and sound waves.

The speed of an electromagnetic wave is given by c = lf. How does the speed depend on frequency? On wavelength?

When astronomers observe a supernova explosion in a distant galaxy, they see a sudden, simultaneous rise in visible light and other forms of electromagnetic radiation. How is this evidence that the speed of light is independent of frequency?

Turning a TV antenna so its rods point vertically may change the quality of your TV reception. Why?

The Sun emits about half of its electromagnetic-wave energy in the visible region of the spectrum. Where do you think it emits most of the remainder?

An LC circuit is made entirely from superconducting materials, yet its oscillations eventually damp out. Why?

If you double the field strength in an electromagnetic wave, what happens to the intensity?

The intensity of light drops as the inverse square of the distance from the source. Does this mean that electromagnetic energy is lost? Explain.

Electromagnetic waves dont readily penetrate metals. Why not?

A uniform electric field is increasing at 1.5 1V/m2/s. Find the displacement current through a 1@cm2 area perpendicular to the field.

A parallel-plate capacitor has square plates 10 cm on a side and 0.50 cm apart. The voltage across the plates is increasing at 220 V/ms. Whats the displacement current in the capacitor?

The fields of an electromagnetic wave are E S = Ep sin1kz + vt2nj and B S = Bp sin1kz + vt2ni. Give a unit vector in the waves propagation direction.

A radio waves electric field is given by the expression E S = E sin 1kz - vt2 * 1ni + nj2. (a) Find the peak electric field. (b) Give a unit vector in the direction of the magnetic field at a place and time where sin1kz - vt2 is positive.

A light-minute is the distance light travels in 1 minute. Show that the Sun is about 8 light-minutes from Earth.

Your intercontinental telephone call is carried by electromagnetic waves routed via a satellite in geostationary orbit at 36,000 km altitude. Approximately how long does it take before your voice is heard at the other end?

An airplanes radar altimeter works by bouncing radio waves off the ground and measuring the round-trip travel time. If that time is 74.7s, what should the pilot report to the passengers as the current altitude?

Roughly how long does it take light to travel 1 foot?

If you speak via radio from Earth to an astronaut on the Moon, how long is it before you can receive a reply?

What are the wavelengths of (a) a 100-MHz FM radio wave, (b) a 5.0-GHz WiFi signal, (c) a 600-THz light wave, and (d) a 1.0-EHz X ray?

A 60-Hz power line emits electromagnetic radiation. Whats the wavelength?

Microwave ovens for consumers use operate at 2.45 GHz. Whats the distance between wave crests in such a microwave?

An electromagnetic wave is propagating in the z-direction. Whats its polarization direction if its magnetic field is in the y-direction?

Polarized light is incident on a sheet of polarizing material, and only 20% of the light gets through. Find the angle between the electric field and the materials transmission axis.

Vertically polarized light passes through a polarizer with its axis at 70 to the vertical. What fraction of the incident intensity emerges from the polarizer?

A typical laboratory electric field is 1500 V/m. Find the average intensity of an electromagnetic wave with this value for its peak field.

What would be the average intensity of a laser beam so strong that its electric field produced dielectric breakdown of air (which requires Ep = 3 MV/m)?

Estimate the peak electric field inside a 1.1-kW microwave oven under the simplifying approximation that the microwaves propagate as a plane wave through the ovens 750@cm2 cross-sectional area.

Your new radio says it can pick up signals with peak electric fields as weak as 450V/m. Will it work if you take it to your remote cabin, where the intensity of your favorite radio station is 0.35nW/m2 ?

A laser pointer delivers 0.10-mW average power in a beam 0.90 mm in diameter. Find (a) the average intensity, (b) the peak electric field, and (c) the peak magnetic field.

Your university radio station has a 5.0-kW radio transmitter that broadcasts uniformly in all directions; listeners within 15 km have reliable reception. You want to increase the power to double that range. What should be the new power?

A parallel-plate capacitor has circular plates with radius 50.0 cm and spacing 1.0 mm. A uniform electric field between the plates is changing at the rate of 1.0 MV/m #s. Find the magnetic field between the plates (a) on the symmetry axis, (b) 15 cm from the axis, and (c) 150 cm from the axis.

Youre engineering a new cell phone, and youd like to incorporate the antenna entirely within the phone, which is 9 cm long when closed. The antenna is to be a quarter-wavelength longa common design for vertically oriented antennas. If the cell-phone frequency is 2.4 GHz, will the antenna fit?

An electric field points into the page and occupies a circular region of radius 1.0 m, as shown in Fig. 29.14. There are no electric charges in the region, but there is a magnetic field forming closed loops pointing clockwise, as shown. The magnetic-field strength 50 cm from the center of the region is 2.0T. (a) Whats the rate of change of the electric field? (b) Is the electric field increasing or decreasing?

The medical profession divides the ultraviolet region of the electromagnetic spectrum into three bands: UVA (320 nm420 nm), UVB (290 nm320 nm), and UVC (100 nm290 nm). UVA and UVB promote skin cancer and premature skin aging; UVB also causes sunburn, but helpfully fosters production of vitamin D. Ozone in Earths atmosphere blocks most of the more dangerous UVC. Find the frequency range associated with UVB radiation.

Dielectric breakdown in air occurs when the electric field is approximately 3 MV/m. What would be the peak magnetic field in an electromagnetic wave with this peak electric field?

A radio receiver can detect signals with electric fields as weak as 320V/m. Find the corresponding magnetic field.

A polarizer blocks 75% of a polarized light beam. Whats the angle between the beams polarization and the polarizers axis?

An electro-optic modulator is a device that switches a laser beam rapidly from off to on by switching the polarization direction through 90 when a voltage is applied. But a brownout results in only enough voltage for a 72 rotation. What fraction of the light is transmitted during the brownout when the beam is supposed to be fully on?

Unpolarized light of intensity S0 passes first through a polarizer with its axis vertical and then through one with its axis at 35 to the vertical. Find the intensity after the second polarizer.

Vertically polarized light passes through two polarizers, the first at 60 to the vertical and the second at 90 to the vertical. What fraction of the light gets through?

Show that its impossible for an electromagnetic wave in vacuum to have a time-varying component of its electric field in the direction of its magnetic field. (Hint: Assume E S does have such a component, and show that you cant satisfy both Gausss and Faradays laws.)

High microwave intensities can cause biological damage through heating of tissue; a particular concern is cataract formation. The U.S. Food and Drug Administration limits microwave radiation near the door of a microwave oven to 5.0 mW/m2 . The window in a particular oven door measures 40 cm by 17 cm and is covered with a metal screen to block microwaves. Assuming power leaks uniformly through the window area, what percent of the ovens 900-W microwave power can leak without exceeding the FDA standards?

Use the fact that sunlight intensity at Earths orbit is 1364 W/m2 to calculate the Suns total power output.

A quasar 10 billion light-years from Earth appears the same brightness as a star 50,000 light-years away. How do the power outputs of quasar and star compare?

Lasers are classified according to the eye-damage danger they pose. Class 2 lasers, including many laser pointers, produce visible light with no greater than 1 mW total power. Theyre relatively safe because the eyes blink reflex limits exposure time to 250 ms. Find (a) the intensity of a 1-mW class 2 laser with beam diameter 1.0 mm, (b) the total energy delivered before the blink reflex shuts the eye, and (c) the peak electric field in the laser beam.

At 1.5 km from a radio transmitter, the peak electric field is 350 mV/m. Assuming the transmitter broadcasts equally in all directions, find (a) the transmitted power and (b) the peak electric field 10 km from the transmitter.

Find the peak electric and magnetic fields 1.5 m from a 60-W lightbulb that radiates equally in all directions.

A typical fluorescent lamp is a little more than 1 m long and a few cm in diameter. How do you expect the light intensity to vary with distance (a) near the lamp but not near either end and (b) far from the lamp?

A camera flash delivers 2.5 kW of light power for 1.0 ms. Find (a) the total energy and (b) the total momentum carried by the flash.

A laser produces an average power of 7.0 W in a 1.0-mm-diameter beam. Find (a) the average intensity and (b) the peak electric field of the laser light.

A 180@W/cm2 laser beam shines on a light-absorbing surface. Whats the radiation pressure on the surface?

A 65-kg astronaut is floating in empty space. If she shines a 1.0-W flashlight in a fixed direction, how long will it take her to accelerate to 10 m/s?

A photon rocket emits a beam of light instead of hot gas. How powerful a beam would be needed to equal the 40-MN thrust of NASAs new Space Launch System (SLS)? Compare your answer with humanitys total electric power-generating capability, about 2 TW.

A white dwarf star is approximately the size of Earth but radiates about as much power as the Sun. Estimate the radiation pressure on a light-absorbing object at the white dwarfs surface.

Use appropriate data from Appendix E to calculate the radiation pressure on a light-absorbing object at the Suns surface.

A radar system produces pulses consisting of exactly 100 full cycles of a sinusoidal 72.5-GHz electromagnetic wave. The average power while the transmitter is on is 66.0 MW, and the waves are confined to a beam 22.4 cm in diameter. Find (a) the peak electric field, (b) the wavelength, (c) the total energy in one pulse, and (d) the total momentum in one pulse. (e) If the transmitter produces 945 pulses each second, whats its average power output?

A cylindrical resistor of length L, radius a, and resistance R carries current I. Calculate the electric and magnetic fields at the surface of the resistor, assuming the electric field is uniform over the surface. Calculate the Poynting vector and show that it points into the resistor. Calculate the flux of the Poynting vector (that is, 1 S S# dA S ) over the resistors surface to get the rate of electromagnetic energy flow into the resistor, and show that the result is I 2 R. Your result shows that the energy heating the resistor comes from the fields surrounding it. These fields are sustained by the source of electric energy that drives the current.

In a stack of polarizing sheets, each sheet has its transmission axis rotated 14 with respect to the preceding sheet. If the stack passes 37% of the incident unpolarized light, how many sheets does it contain?

Youre an astronomer studying the origin of the solar system, and youre evaluating a hypothesis that sufficiently small particles were blown out of the solar system by the force of sunlight. To see how small such particles must be, compare the force of sunlight with the force of solar gravity, and solve for the particle radius at which the two are equal. Assume spherical particles with density 2 g/cm3 . (Note: Distance from the Sun doesnt matter. Why not?)

Differentiate Equation 29.12 with respect to x and Equation 29.13 with respect to t. Then, using the fact that mixed derivatives are equal (e.g., 0 0t a 0B dx b = 0 0x a 0B dt b ), combine the resulting equations and show that the result is the wave equation (Equation 14.5) for waves with speed c = 1/2P0m0.

Maxwells equations in a dielectric resemble those in vacuum (Equations 29.629.9) but with P0 replaced by kP0, where k is the dielectric constant introduced in Chapter 23. Show that the speed of electromagnetic waves in a dielectric is c/1k

A friend buys a used pickup truck that comes with a CB radio. However, the antenna is broken off, and your friend asks you to help make one out of a steel rod that he will affix to the rear bumper. You know that the CB channel frequency is 27.3 MHz and that the antenna must be a quarter-wavelength long. How long should you make the rod?

Your roommates father is CEO of a coal company, so your roommate is understandably skeptical of alternative energy proposals. He claims that theres no future for solar energy, because the power in sunlight is insufficient to meet humankinds energy demand. Is he right? To find out, compare the solar power incident on Earth with our human energy consumption rate of about 16 TW.

The Voyager 1 spacecraft is now beyond the outer reaches of our solar system, but earthbound scientists still receive data from the spacecrafts 20-W radio transmitter. Voyager is expected to continue transmitting until about 2025, when it will be some 25 billion km from Earth. Whats the diameter of a dish antenna that will receive 10-20 W of power from Voyager at this time?

Your friend who works for the college radio station must make electric-field measurements for a report to be filed with the stations application for license renewal. The measurement is made 4.6 km from the antenna, where your friend measures the electric field at 380 V/m. The station is allowed to broadcast at no more than 55-kW power. Assuming power spreads equally in all directions, is the station in compliance with its license?

The National Ignition Facility at Lawrence Livermore National Laboratory initiates nuclear fusion by converging 192 laser beams on a deuteriumtritium target. Each beam has a square cross section 38 cm on a side, and each beam delivers 10.0 kJ of energy in 20.0 ns. Find (a) the peak electric field and (b) the peak magnetic field in each laser beam. (c) Find the combined power of all 192 laser beams while theyre firing, and compare with humankinds energy consumption rate of about 16 TW

The table below shows the intensity of the radio signal received at Earth from a spacecraft on its way to the outer solar system, as a function of its distance from Earth. Distances are in astronomical units (AU, with 1 AU being the mean EarthSun distance; see Appendix E). Determine a quantity that, when you plot S versus that quantity, should give a straight line. Make your plot, establish a best-fit line, and use it to determine the spacecrafts transmitter power.

Proposals have been made to sail spacecraft to the outer solar system using the pressure of sunlight, or even to propel interstellar spacecraft with high-powered, Earth-based lasers. Sailing spacecraft would need no fuela great advantage because fuel constitutes much of the initial weight of any space mission. NASAs Sunjammer, launched in 2015, is the first sailing spacecraft deployed beyond Earth orbit. Figure 29.15 shows what a similar solar sail might look like in space. If a sunlight-powered sailing spacecraft accelerated at 1m/s2 in the vicinity of Earths orbit, what would be its acceleration at Mars, about 1.5 times as far from the Sun as Earth? a. about 0.25 m/s2 b. a little less than 0.5 m/s2 c. a little more than 0.5 m/s2 d. about 0.66 m/s2

Proposals have been made to sail spacecraft to the outer solar system using the pressure of sunlight, or even to propel interstellar spacecraft with high-powered, Earth-based lasers. Sailing spacecraft would need no fuela great advantage because fuel constitutes much of the initial weight of any space mission. NASAs Sunjammer, launched in 2015, is the first sailing spacecraft deployed beyond Earth orbit. Figure 29.15 shows what a similar solar sail might look like in space. One spacecraft has a sail that absorbs all light incident on it; the other has a perfectly reflective sail. How do their accelerations compare in light with the same intensity? a. The absorptive sail gives twice the acceleration. b. The reflective sail gives twice the acceleration. c. The absorptive sail gives greater acceleration, but not twice as much. d. The reflective sail gives greater acceleration, but not twice as much.

Proposals have been made to sail spacecraft to the outer solar system using the pressure of sunlight, or even to propel interstellar spacecraft with high-powered, Earth-based lasers. Sailing spacecraft would need no fuela great advantage because fuel constitutes much of the initial weight of any space mission. NASAs Sunjammer, launched in 2015, is the first sailing spacecraft deployed beyond Earth orbit. Figure 29.15 shows what a similar solar sail might look like in space. A sail capable of propelling a spacecraft to the outer solar system must be able to overcome the Suns gravity. Suppose a spacecraft is designed so the force of sunlight on its sail is 20 times that of solar gravity in the vicinity of Earths orbit. If the spacecraft reaches Jupiter, some 5 times as far from the Sun as Earth, a. the sail force will still exceed solar gravity, now by a factor of 4. b. the sail force will be slightly less than solar gravity. c. the sail force will now be 25 times solar gravity. d. the sail force will still be 20 times solar gravity.

Proposals have been made to sail spacecraft to the outer solar system using the pressure of sunlight, or even to propel interstellar spacecraft with high-powered, Earth-based lasers. Sailing spacecraft would need no fuela great advantage because fuel constitutes much of the initial weight of any space mission. NASAs Sunjammer, launched in 2015, is the first sailing spacecraft deployed beyond Earth orbit. Figure 29.15 shows what a similar solar sail might look like in space. The intensity of sunlight at Earths orbit is about 1.4 kW/m2 . A 100-kg sailing spacecraft with 1@km2 sail area would experience an acceleration of about a. 5 mm/s2 . b. 5 cm/s2 . c. 5 m/s2 . d. 5 km/s2 .