|| What is the electric field at the position of the electron in FIGURE P31.27? Give your answer in component form.

FIGURE P31.27

|| What is the force (magnitude and direction) on the proton in FIGURE P31.28? Give the direction as an angle cw or ccw from vertical.

FIGURE P31.28

| A proton is fired with a speed of $1.0\times {10}^6\text{m}\text{/}\text{s}$ through the parallel-plate capacitor shown in FIGURE P31.29. The capacitor’s electric field is $\vec{E}=(1.0\times {10}^5\text{V}\text{/}\text{m}\text{,}\mathrm{down})$.

1. What magnetic field $\vec{B}$, both strength and direction, must be applied to allow the proton to pass through the capacitor with no change in speed or direction?

2. Find the electric and magnetic fields in the proton’s reference frame.

3. How does an experimenter in the proton’s frame explain that the proton experiences no force as the charged plates fly by?

FIGURE P31.29

| What electric field strength and direction will allow the proton in FIGURE P31.30 to pass through this region of space without being deflected?

FIGURE P31.30

||| An electron travels with $\vec{v}=5.0\times {10}^6î\text{m}\text{/}\text{s}$ through a point in space where $\vec{E}=(2.0\times {10}^5î-2.0\times {10}^5ĵ)\text{V}\text{/}\text{m}$ and $\vec{B}=-0.10\widehat{k}\text{T}$. What is the force on the electron?

|| INT In FIGURE P31.32, a circular loop of radius r travels with speed v along a charged wire having linear charge density $\text{λ}$. The wire is at rest in the laboratory frame, and it passes through the center of the loop.

1. What are $\vec{E}$ and $\vec{B}$ at a point on the loop as measured by a scientist in the laboratory? Include both strength and direction.

2. What are the fields $\vec{E}$ and $\vec{B}$ at a point on the loop as measured by a scientist in the frame of the loop?

3. Show that an experimenter in the loop’s frame sees a current $I=\text{λ}v$ passing through the center of the loop.

4. What electric and magnetic fields would an experimenter in the loop’s frame calculate at distance r from the current of part c?

5. Show that your fields of parts b and d are the same.

FIGURE P31.32

|| INT The magnetic field inside a 4.0-cm-diameter superconducting solenoid varies sinusoidally between 8.0 T and 12.0 T at a frequency of 10 Hz.

1. What is the maximum electric field strength at a point 1.5 cm from the solenoid axis?

2. What is the value of B at the instant E reaches its maximum value?

|| INT A simple series circuit consists of a $150\text{Ω}$ resistor, a 25 V battery, a switch, and a 2.5 pF parallel-plate capacitor (initially uncharged) with plates 5.0 mm apart. The switch is closed at $t=0\text{s}$.

1. After the switch is closed, find the maximum electric flux and the maximum displacement current through the capacitor.

2. Find the electric flux and the displacement current at $t=0.50\text{ns}$.

|| A wire with conductivity $\sigma$ carries current I. The current is increasing at the rate $dI\text{/}dt$.

1. Show that there is a displacement current in the wire equal to $\left({ϵ}_0\text{/}\sigma \right)\left(dI\text{/}dt\right)$.

2. Evaluate the displacement current for a copper wire in which the current is increasing at $1.0\times {10}^6\text{A}\text{/}\text{s}$.

|| FIGURE P31.36 shows the voltage across a $0.10\mu \text{F}$ capacitor. Draw a graph showing the displacement current through the capacitor as a function of time.

FIGURE P31.36

|| A 10 A current is charging a 1.0-cm-diameter parallel-plate capacitor.

1. What is the magnetic field strength at a point 2.0 mm radially from the center of the wire leading to the capacitor?

2. What is the magnetic field strength at a point 2.0 mm radially from the center of the capacitor?

|| FIGURE P31.38 shows the electric field inside a cylinder of radius $R=3.0\text{mm}$. The field strength is increasing with time as $E=1.0\times {10}^8{t}^2\text{V}\text{/}\text{m}$, where t is in s. The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for $t<0$.

1. Find an expression for the electric flux ${\text{Φ}}_{\text{e}}$ through the entire cylinder as a function of time.

2. Draw a picture showing the magnetic field lines inside and outside the cylinder. Be sure to include arrowheads showing the field’s direction.

3. Find an expression for the magnetic field strength as a function of time at a distance $r<R$ from the center. Evaluate the magnetic field strength at $r=2.0\text{mm, t}=2.0s$.

4. Find an expression for the magnetic field strength as a function of time at a distance $r>R$ from the center. Evaluate the magnetic field strength at $r=4.0\text{mm, t}=2.0\text{s}$.

FIGURE P31.38

|| A long, thin superconducting wire carrying a 15 A current passes through the center of a thin, 2.0-cm-diameter ring. A uniform electric field of increasing strength also passes through the ring, parallel to the wire. The magnetic field inside the ring is zero.

1. At what rate is the electric field strength increasing?

2. Is the electric field in the direction of the current or opposite to the current?

|| A $1.0\mu \text{F}$ capacitor is discharged, starting at $t=0\text{s}$.The displacement current between the plates is $I_{\text{disp}}=$ $(10\text{A)exp}(-t\text{/}2.0\mu \text{s})$. What was the capacitor’s initial voltage $(\text{Δ}{V}_{\text{C}}){}_0?$

|| At one instant, the electric and magnetic fields at one point of an electromagnetic wave are $\vec{E}=(200î+300ĵ-50\widehat{k})\text{V}\text{/}\text{m}$ and $\vec{B}=B_0(7.3î-7.3ĵ+a\widehat{k})\mu \text{T}$.

1. What are the values of a and $B_0?$

2. What is the Poynting vector at this time and position?

||

1. Show that $u_{\text{E}}$ and $u_{\text{B}}$, the energy densities of the electric and magnetic fields, are equal to each other in an electromagnetic wave. In other words, show that the wave’s energy is divided equally between the electric field and the magnetic field.

2. What is the total energy density in an electromagnetic wave of intensity $1000\text{W}\text{/}{\text{m}}^2?$

| The intensity of sunlight reaching the earth is $1360\text{W}\text{/}{\text{m}}^2$.

1. What is the power output of the sun?

2. What is the intensity of sunlight reaching Mars?

| Assume that a 7.0-cm-diameter, 100 W lightbulb radiates all its energy as a single wavelength of visible light. Estimate the electric and magnetic field amplitudes at the surface of the bulb.

|| An electromagnetic wave has an electric field amplitude of $50\text{V}/\text{m}$. The field has a maximum rate of change of $7.5\times {10}^{10}\left(\text{V}\text{/}\text{m}\right)\text{/}\text{s}$. What is the wavelength?

| When the Voyager 2 spacecraft passed Neptune in 1989, it was $4.5\times {10}^9\text{km}$ from the earth. Its radio transmitter, with which it sent back data and images, broadcast with a mere 21 W of power. Assuming that the transmitter broadcast equally in all directions,

1. What signal intensity was received on the earth?

2. What electric field amplitude was detected?

The received signal was somewhat stronger than your result because the spacecraft used a directional antenna, but not by much.

|| In reading the instruction manual that came with your garage-door opener, you see that the transmitter unit in your car produces a 250 mW, 350 MHz signal and that the receiver unit is supposed to respond to a radio wave at this frequency if the electric field amplitude exceeds $0.10\text{V}\text{/}\text{m}$. You wonder if this is really true. To find out, you put fresh batteries in the transmitter and start walking away from your garage while opening and closing the door. Your garage door finally fails to respond when you’re 42 m away. What is the electric field amplitude at the receiver when it first fails?

|| The maximum electric field strength in air is $3.0\text{MV}\text{/}\text{m}$. Stronger electric fields ionize the air and create a spark. What is the maximum power that can be delivered by a 1.0-cm-diameter laser beam propagating through air?

|| BIO A LASIK vision-correction system uses a laser that emits 10-ns-long pulses of light, each with 2.5 mJ of energy. The laser beam is focused to a 0.85-mm-diameter circle on the cornea. What is the electric field amplitude of the light wave at the cornea?

|| The intensity of sunlight reaching the earth is $1360\text{W}\text{/}{\text{m}}^2$. Assuming all the sunlight is absorbed, what is the radiation-pressure force on the earth? Give your answer (a) in newtons and (b) as a fraction of the sun’s gravitational force on the earth.

|| BIO INT For radio and microwaves, the depth of penetration into the human body is proportional to ${\text{λ}}^{1\text{/}2}$. If 27 MHz radio waves penetrate to a depth of 14 cm, how far do 2.4 GHz microwaves penetrate?

|| A laser beam shines straight up onto a flat, black foil of mass m.

1. Find an expression for the laser power P needed to levitate the foil.

2. Evaluate P for a foil with a mass of $25\mu \text{g}$.

| For a science project, you would like to horizontally suspend an 8.5 by 11 inch sheet of black paper in a vertical beam of light whose dimensions exactly match the paper. If the mass of the sheet is 1.0 g, what light intensity will you need?

|| INT Large quantities of dust should have been left behind after the creation of the solar system. Larger dust particles, comparable in size to soot and sand grains, are common. They create shooting stars when they collide with the earth’s atmosphere. But very small dust particles are conspicuously absent. Astronomers believe that the very small dust particles have been blown out of the solar system by the sun. By comparing the forces on dust particles, determine the diameter of the smallest dust particles that can remain in the solar system over long periods of time. Assume that the dust particles are spherical, black, and have a density of $2000\text{kg}\text{/}{\text{m}}^3$. The sun emits electromagnetic radiation with power $3.9\times {10}^{26}\text{W}$.

|| INT You’ve recently read about a chemical laser that generates a 20-cm-diameter, 25 MW laser beam. One day, after physics class, you start to wonder if you could use the radiation pressure from this laser beam to launch small payloads into orbit. To see if this might be feasible, you do a quick calculation of the acceleration of a 20-cm-diameter, 100 kg, perfectly absorbing block. What speed would such a block have if pushed horizontally 100 m along a frictionless track by such a laser?

|| Unpolarized light of intensity $I_0$ is incident on three polarizing filters. The axis of the first is vertical, that of the second is $45\text{°}$ from vertical, and that of the third is horizontal. What light intensity emerges from the third filter?

|| Unpolarized light of intensity $I_0$ is incident on two polarizing filters. The transmitted light intensity is $I_0/10$. What is the angle between the axes of the two filters?

|| Unpolarized light of intensity $I_0$ is incident on a stack of 7 polarizing filters, each with its axis rotated $15\text{°}$ cw with respect to the previous filter. What light intensity emerges from the last filter?

|| INT It has been proposed that small spacecraft could reach other planets in a fairly short time—days instead of many months—if they unfurl a reflective sail and are accelerated by a powerful laser beam generated by an earth-orbiting laser.

1. What speed would a spacecraft need to reach Mars in 8.0 days when Mars is closest to earth?

2. The laser beam would need to stay focused on the spacecraft’s sail during the acceleration phase. Laser beams slowly increase in diameter due to diffraction—the topic of Chapter 33—but it’s possible that an extension of current technology would allow the diameter of a laser beam to be no more than 100 m, a plausible diameter for a sail, at a distance 10 times the distance to the moon. What steady acceleration would a spacecraft need to reach the speed you found in part a in a distance 10 times that from the earth to the moon?

3. For how long, in hours, would the acceleration need to last?

4. What laser beam power, in GW, would be needed to send a 10 kg probe to Mars in 8.0 days? (That probe would be large enough to carry some sensors, cameras, and communication equipment.) Your answer is about a factor of 5000 greater than the most powerful continuous laser developed to date. It would be quite an engineering challenge to build such a laser, but there’s no inherent reason it couldn’t be done.

| The concept of space-based solar power was proposed 50 years ago. It is technically feasible but so far not economically feasible. The idea is that a giant solar-panel array in geosynchronous orbit would collect solar energy, convert the energy to microwaves, and then beam the microwaves down to a large antenna on the ground below. The efficiency of converting solar energy to electricity and to microwaves is about 30%; the efficiency of converting microwaves to electric power on the grid is 85%. The somewhat low efficiency is offset by the fact that, unlike terrestrial solar panels, the satellite would be continuously in sunlight and could provide solar energy 24 hours a day.

1. A large nuclear or fossil-fuel power plant generates 1.0 GW of electric power. To replace this power plant with space-based solar power, what would the diameter of a circular solar-cell array need to be? The intensity of sunlight above the earth’s atmosphere is $1300{\text{W/m}}^2$. Your answer is large, but still feasible.

2. There is some concern that birds or stray aircraft would be incinerated if they flew through the microwave beam. What is the minimum diameter of a receiving antenna that would keep the electric field amplitude just above the ground from exceeding $1600\text{V}/\text{m}$, a value known to be safe? Your answer is again large, but still feasible.

||| Laboratory scientists have created the electric and magnetic fields shown in FIGURE CP31.61. These fields are also seen by scientists who zoom past in a rocket traveling in the x-direction at $1.0\times {10}^6\text{m}\text{/}\text{s}$. According to the rocket scientists, what angle does the electric field make with the axis of the rocket?

FIGURE CP31.61

||| A cube of water 10 cm on a side is placed in a microwave beam having $E_0=11\text{kV}\text{/}\text{m}$. The microwaves illuminate one face of the cube, and the water absorbs 80% of the incident energy. How long will it take to raise the water temperature by $50\text{°}\text{C}?$ Assume that the water has no heat loss during this time.

||| An 80 kg astronaut has gone outside his space capsule to do some repair work. Unfortunately, he forgot to lock his safety tether in place, and he has drifted 5.0 m away from the capsule. Fortunately, he has a 1000 W portable laser with fresh batteries that will operate it for 1.0 h. His only chance is to accelerate himself toward the space capsule by firing the laser in the opposite direction. He has a 10 h supply of oxygen. How long will it take him to reach safety?

||| An electron travels with $\vec{v}=5.0\times {10}^6î\text{m}\text{/}\text{s}$ through a point in space where $\vec{B}=0.10ĵ\text{T}$. The force on the electron at this point is $\vec{F}=(9.6\times {10}^{-14}î-9.6\times {10}^{-14}\widehat{k})\text{N}$. What is the electric field?

||| The radar system at an airport broadcasts 11 GHz microwaves with 150 kW of power. An approaching airplane with a $31{\text{m}}^2$ cross section is 30 km away. Assume that the radar broadcasts uniformly in all directions and that the airplane scatters microwaves uniformly in all directions. What is the electric field strength of the microwave signal received back at the airport $200\mu \text{s}$ later?

||| CALC Consider current I passing through a resistor of radius r, length L, and resistance R.

1. Determine the electric and magnetic fields at the surface of the resistor. Assume that the electric field is uniform throughout, including at the surface.

2. Determine the strength and direction of the Poynting vector at the surface of the resistor.

3. Show that the flux of the Poynting vector (i.e., the integral of $\vec{S}\cdot d\vec{A})$ over the surface of the resistor is $I^{2}R$. Then give an interpretation of this result.