Solution: A radio station is allowed to broadcast at an

Determine the rate at which the electric field changes between the round plates of a capacitor, 6.0 cm in diameter, if the plates are spaced 1.1 mm apart and the voltage across them is changing at a rate of 120 V/s.

Calculate the displacement current /D between the square plates, 5.8 cm on a side, of a capacitor if the electric field is changing at a rate of 2.0 X 106 V/m s.

At a given instant, a 2.8-A current flows in the wires connected to a parallel-plate capacitor. What is the rate at which the electric field is changing between the plates if the square plates are 1.60 cm on a side?

A 1500-nF capacitor with circular parallel plates 2.0 cm in diameter is accumulating charge at the rate of 38.0 mC/s at some instant in time. What will be the induced magnetic field strength 10.0 cm radially outward from the center of the plates? What will be the value of the field strength after the capacitor is fully charged?

Show that the displacement current through a parallelplate capacitor can be written /D = C dV/dt, where V is the voltage across the capacitor at any instant.

Suppose an air-gap capacitor has circular plates of radius R = 2.5 cm and separation d = 1.6 mm. A 76.0-Hz emf, % = fo cos (tit, is applied to the capacitor. The maximum displacement current is 35 /iA . Determine (a) the maximum conduction current I, (b) the value of %, (c) the maximum value of d<$>E/dt between the plates. Neglect fringing.

Suppose that a circular parallel-plate capacitor has radius R0 = 3.0 cm and plate separation d = 5.0 mm. A sinusoidal potential difference V = V0 sin (2-77ft) is applied across the plates, where Vq = 150 V and / = 60 Hz. (a) In the region between the plates, show that the magnitude of the induced magnetic field is given by B = B0(R) cos(2ttft), where R is the radial distance from the capacitors central axis. (b) Determine the expression for the amplitude B0(R) of this time-dependent (sinusoidal) field when R < R0, and when R > R0. (c) Plot B0(R) in tesla for the range 0 < R < 10 cm.

If the electric field in an EM wave has a peak magnitude of 0.57 X 10_4V/m, what is the peak magnitude of the magnetic field strength?

If the magnetic field in a traveling EM wave has a peak magnitude of 12.5 nT, what is the peak magnitude of the electric field?

In an EM wave traveling west, the B field oscillates vertically and has a frequency of 80.0 kHz and an rms strength of 7.75 X 10_9T. Determine the frequency and rms strength of the electric field. What is its direction?

The electric field of a plane EM wave is given by Ex = E0 cos(kz + (tit), Ey = Ez = 0. Determine (a) the direction of propagation and (b) the magnitude and direction of B.

Consider two possible candidates E(x,t) as solutions of the wave equation for an EM waves electric field. Let A and a be constants. Show that (a) E(x,t) = Ae~a^x~vt^2 satisfies the wave equation, and that (b) E(x,t) = A e~ ^~ vt^ does not satisfy the wave equation.

What is the frequency of a microwave whose wavelength is 1.50 cm?

(a) What is the wavelength of a 25.75 X 109Hz radar signal? (b) What is the frequency of an X-ray with wavelength 0.12 nm?

How long does it take light to reach us from the Sun, 1.50 X 108 km away?

An EM wave has frequency 8.56 X 1014 Hz. What is its wavelength, and how would we classify it?

Electromagnetic waves and sound waves can have the same frequency, (a) What is the wavelength of a 1.00-kHz electromagnetic wave? (b) What is the wavelength of a 1.00-kHz sound wave? (The speed of sound in air is 341 m/s.) (c) Can you hear a 1.00-kHz electromagnetic wave?

Pulsed lasers used for science and medicine produce very brief bursts of electromagnetic energy. If the laser light wavelength is 1062 nm (Neodymium-YAG laser), and the pulse lasts for 38 picoseconds, how many wavelengths are found within the laser pulse? How brief would the pulse need to be to fit only one wavelength?

How long would it take a message sent as radio waves from Earth to reach Mars (a) when nearest Earth, (b) when farthest from Earth?

An electromagnetic wave has an electric field given by E = i(225 V/m) sin[(0.077 m-1)z (2.3 X 107rad/s)?]. (a) What are the wavelength and frequency of the wave? (b) Write down an expression for the magnetic field.

What is the minimum angular speed at which Michelsons eight-sided mirror would have had to rotate to reflect light into an observers eye by succeeding mirror faces (1/8 of a revolution, Fig. 31-14)?

The E field in an EM wave has a peak of 26.5 mV/m. What is the average rate at which this wave carries energy across unit area per unit time?

The magnetic field in a traveling EM wave has an rms strength of 22.5 nT. How long does it take to deliver 335 J of energy to 1.00 cm2 of a wall that it hits perpendicularly?

How much energy is transported across a 1.00 cm2 area per hour by an EM wave whose E field has an rms strength of 32.8 mV/m?

A spherically spreading EM wave comes from a 1500-W source. At a distance of 5.0 m, what is the intensity, and what is the rms value of the electric field?

If the amplitude of the B field of an EM wave is 2.5 X 10- 7T, (a) what is the amplitude of the E field? (b) What is the average power per unit area of the EM wave?

What is the energy contained in a 1.00-m3 volume near the Earths surface due to radiant energy from the Sun? See Example 31-6.

A 15.8-mW laser puts out a narrow beam 2.00 mm in diameter. What are the rms values of E and B in the beam?

Estimate the average power output of the Sun, given that about 1350 W/m2 reaches the upper atmosphere of the Earth.

A high-energy pulsed laser emits a 1.0-ns-long pulse of average power 1.8 X 1011W. The beam is 2.2 X 10_3m in radius. Determine (a) the energy delivered in each pulse, and (b) the rms value of the electric field.

How practical is solar power for various devices? Assume that on a sunny day, sunlight has an intensity of 1000 W/m2 at the surface of Earth and that, when illuminated by that sunlight, a solar-cell panel can convert 10% of the sunlights energy into electric power. For each device given below, calculate the area A of solar panel needed to power it. (a) A calculator consumes 50 mW. Find A in cm2. Is A small enough so that the solar panel can be mounted directly on the calculator that it is powering? (b) A hair dryer consumes 1500 W. Find A in m2. Assuming no other electronic devices are operating within a house at the same time, is A small enough so that the hair dryer can be powered by a solar panel mounted on the houses roof? (c) A car requires 20 hp for highway driving at constant velocity (this car would perform poorly in situations requiring acceleration). Find A in m2. Is A small enough so that this solar panel can be mounted directly on the car and power it in real time?

(a) Show that the Poynting vector S points radially inward toward the center of a circular parallel-plate capacitor when it is being charged as in Example 31-1. (b) Integrate S over the cylindrical boundary of the capacitor gap to show that the rate at which energy enters the capacitor is equal to the rate at which electrostatic energy is being stored in the electric field of the capacitor (Section 24-4). Ignore fringing of E.

The Arecibo radio telescope in Puerto Rico can detect a radio wave with an intensity as low as 1 X 10_23W/m2. As a best-case scenario for communication with extraterrestrials, consider the following: suppose an advanced civilization located at point A, a distance x away from Earth, is somehow able to harness the entire power output of a Sun-like star, converting that power completely into a radio-wave signal which is transmitted uniformly in all directions from A. (a) In order for Arecibo to detect this radio signal, what is the maximum value for x in light-years ( l l y ~ 1 0 16m)? (b) How does this maximum value compare with the 100,000-ly size of our Milky Way galaxy? The intensity of sunlight at Earths orbital distance from the Sun is 1350 W/m2.

Estimate the radiation pressure due to a 75-W bulb at a distance of 8.0 cm from the center of the bulb. Estimate the force exerted on your fingertip if you place it at this point.

Laser light can be focused (at best) to a spot with a radius r equal to its wavelength A. Suppose that a 1.0-W beam of green laser light (A = 5 X 10-7 m) is used to form such a spot and that a cylindrical particle of about that size (let the radius and height equal r) is illuminated by the laser as shown in Fig. 31-23. Estimate the acceleration of the particle, if its density equals that of water and it absorbs the radiation. [This order-of-magnitude calculation convinced researchers of the feasibility of optical tweezers, p. 829.]FIGURE 31-23 Problem 35.

The powerful laser used in a laser light show provides a 3-mm diameter beam of green light with a power of 3 W. When a space-walking astronaut is outside the Space Shuttle, her colleague inside the Shuttle playfully aims such a laser beam at the astronauts space suit. The masses of the suited astronaut and the Space Shuttle are 120 kg and 103,000 kg, respectively, (a) Assuming the suit is perfectly reflecting, determine the radiation-pressure force exerted on the astronaut by the laser beam, (b) Assuming the astronaut is separated from the Shuttles center of mass by 20 m, model the Shuttle as a sphere in order to estimate the gravitation force it exerts on the astronaut, (c) Which of the two forces is larger, and by what factor?

What size should the solar panel on a satellite orbiting Jupiter be if it is to collect the same amount of radiation from the Sun as a 1.0-m2 solar panel on a satellite orbiting Earth?

What is the range of wavelengths for (a) FM radio (88 MHz to 108 MHz) and (b) AM radio (535 kHz to 1700 kHz)?

Estimate the wavelength for 1.9-GHz cell phone reception.

The variable capacitor in the tuner of an AM radio has a capacitance of 2200 pF when the radio is tuned to a station at 550 kHz. What must the capacitance be for a station near the other end of the dial, 1610 kHz?

A certain FM radio tuning circuit has a fixed capacitor C = 620 pF. Tuning is done by a variable inductance. What range of values must the inductance have to tune stations from 88 MHz to 108 MHz?

A satellite beams microwave radiation with a power of 12 kW toward the Earths surface, 550 km away. When the beam strikes Earth, its circular diameter is about 1500 m. Find the rms electric field strength of the beam at the surface of the Earth.

A 1.60-m-long FM antenna is oriented parallel to the electric field of an EM wave. How large must the electric field be to produce a 1.00-mV (rms) voltage between the ends of the antenna? What is the rate of energy transport per square meter?

Who will hear the voice of a singer first: a person in the balcony 50.0 m away from the stage (see Fig. 31-24), or a person 1500 km away at home whose ear is next to the radio listening to a live broadcast? Roughly how much sooner? Assume the microphone is a few centimeters from the singer and the temperature is 20 C.FIGURE 31-24 Problem 44.

Light is emitted from an ordinary lightbulb filament in wave-train bursts about 10_8s in duration. What is the length in space of such wave trains?

Radio-controlled clocks throughout the United States receive a radio signal from a transmitter in Fort Collins, Colorado, that accurately (within a microsecond) marks the beginning of each minute. A slight delay, however, is introduced because this signal must travel from the transmitter to the clocks. Assuming Fort Collins is no more than 3000 km from any point in the U.S., what is the longest travel-time delay?

A radio voice signal from the Apollo crew on the Moon (Fig. 31-25) was beamed to a listening crowd from a radio speaker. If you were standing 25 m from the loudspeaker, what was the total time lag between when you heard the sound and when the sound entered a microphone on the Moon and traveled to Earth? FIGURE 31-25 Problem 47.

Cosmic microwave background radiation fills all space with an average energy density of 4 X 10_14J/m3. (a) Find the rms value of the electric field associated with this radiation. (b) How far from a 7.5-kW radio transmitter emitting uniformly in all directions would you find a comparable value?

What are E0 and B0 2.00 m from a 75-W light source? Assume the bulb emits radiation of a single frequency uniformly in all directions.

Estimate the rms electric field in the sunlight that hits Mars, knowing that the Earth receives about 1350 W/m2 and that Mars is 1.52 times farther from the Sun (on average) than is the Earth.

At a given instant in time, a traveling EM wave is noted to have its maximum magnetic field pointing west and its maximum electric field pointing south. In which direction is the wave traveling? If the rate of energy flow is 560 W/m2, what are the maximum values for the two fields?

How large an emf (rms) will be generated in an antenna that consists of a circular coil 2.2 cm in diameter having 320 turns of wire, when an EM wave of frequency 810 kHz transporting energy at an average rate of 1.0 X 10-4 W/m2 passes through it? [Hint, you can use Eq. 29-4 for a generator, since it could be applied to an observer moving with the coil so that the magnetic field is oscillating with the frequency / = a)/2it.\

The average intensity of a particular TV stations signal is 1.0 X 10-13 W/m2 when it arrives at a 33-cm-diameter satellite TV antenna, (a) Calculate the total energy received by the antenna during 6.0 hours of viewing this stations programs. (b) What are the amplitudes of the E and B fields of the EM wave?

A radio station is allowed to broadcast at an average power not to exceed 25 kW. If an electric field amplitude of 0.020 V/m is considered to be acceptable for receiving the radio transmission, estimate how many kilometers away you might be able to hear this station.

A point source emits light energy uniformly in all directions at an average rate P0 with a single frequency / . Show that the peak electric field in the wave is given by E . = VqcPq l i r r 2

Suppose a 35-kW radio station emits EM waves uniformly in all directions, (a) How much energy per second crosses a 1.0-m2 area 1.0 km from the transmitting antenna? (ib) What is the rms magnitude of the E field at this point, assuming the station is operating at full power? What is the rms voltage induced in a 1 .0-m-long vertical car antenna (c) 1.0 km away, (d) 50 km away?

What is the maximum power level of a radio station so as to avoid electrical breakdown of air at a distance of 0.50 m from the transmitting antenna? Assume the antenna is a point source. Air breaks down in an electric field of about 3 X 106 V/m.

In free space (vacuum), where the net charge and current flow is zero, the speed of an EM wave is given by v = l/Veo^o- If, instead, an EM wave travels in a nonconducting (dielectric) material with dielectric constant K, then v = l / y l e ^ . For frequencies corresponding to the visible spectrum (near 5 X 1014 Hz), the dielectric constant of water is 1.77. Predict the speed of light in water and compare this value (as a percentage) with the speed of light in a vacuum.

The metal walls of a microwave oven form a cavity of dimensions 37 cm X 37 cm X 20 cm. When 2.45-GHz microwaves are continuously introduced into this cavity, reflection of incident waves from the walls set up standing waves with nodes at the walls. Along the 37-cm dimension of the oven, how many nodes exist (excluding the nodes at the wall) and what is the distance between adjacent nodes? [Because no heating occurs at these nodes, most microwaves rotate food while operating.]

Imagine that a steady current I flows in a straight cylindrical wire of radius R0 and resistivity p. (a) If the current is then changed at a rate di/dt, show that a displacement current /D exists in the wire of magnitude e0p(dl/dt). (b) If the current in a copper wire is changed at the rate of 1.0 A/ms, determine the magnitude of /D. (c) Determine the magnitude of the magnetic field BD (T) created by /d at the surface of a copper wire with R0 = 1.0 mm. Compare (as a ratio) BD with the field created at the surface of the wire by a steady current of 1.0 A.

The electric field of an EM wave pulse traveling along the x axis in free space is given by Ey = E0 exp [ - a 2* 2 - (32t2 + 2a(3xt], where Eq, a, and /3 are positive constants, (a) Is the pulse moving in the H-jic or x direction? (b) Express /3 in terms of a and c (speed of light in free space), (c) Determine the expression for the magnetic field of this EM wave.

Suppose that a right-moving EM wave overlaps with a leftmoving EM wave so that, in a certain region of space, the total electric field in the y direction and magnetic field in the z direction are given by Ey = Eq sin(kx cot) + Eq sin(kx + cot) and Bz = Bq sin(/:x - cot) Bq sin(/:x + cot), (a) Find the mathematical expression that represents the standing electric and magnetic waves in the y and z directions, respectively. (b) Determine the Poynting vector and find the x locations at which it is zero at all times.

The electric and magnetic fields of a certain EM wave in free space are given by E = E 0 sin (A:* - cot) j + Eqcos (kx cot) k and B = i?0co s^ x ~~ <*>t)j - B0sin(kx cot)k. (a) Show that E and B are perpendicular to each other at all times, (b) For this wave, E and B are in a plane parallel to the yz plane. Show that the wave moves in a direction perpendicular to both E and B. (c) At any arbitrary choice of position x and time t, show that the magnitudes of E and B always equal E q and Bq, respectively. (d) At x = 0, draw the orientation of E and B in the yz plane at t = 0. Then qualitatively describe the motion of these vectors in the yz plane as time increases. [Note: The EM wave in this Problem is circularly polarized.]