Chapter 23

\(\bullet\) (a) How much time does it take light to travel from the moon to the earth, a distance of \(384,000 \mathrm{km} ?\) (b) Light from the star Sirius takes 8.61 years to reach the earth. What is the distance to Sirius in kilometers?

\(\bullet\) Consider electromagnetic waves propagating in air. (a) Determine the frequency of a wave with a wavelength of (i) \(5.0 \mathrm{km},\) (ii) \(5.0 \mu \mathrm{m},\) (iii) 5.0 \(\mathrm{nm}\) . (b) What is the wavelength (in meters and nanometers) of (i) gamma rays of frequency \(6.50 \times 10^{21} \mathrm{Hz}\) , (ii) an AM station radio wave of frequency 590 \(\mathrm{kHz} ?\)

\(\bullet\) Most people perceive light having a wavelength between 630 \(\mathrm{nm}\) and 700 \(\mathrm{nm}\) as red and light with a wavelength between 400 \(\mathrm{nm}\) and 440 \(\mathrm{nm}\) as violet. Calculate the approximate frequency ranges for (a) violet light and (b) red light.

\(\bullet\) The electric field of a sinusoidal electromagnetic wave obeys the equation \(E=-(375 \mathrm{V} / \mathrm{m}) \sin [(5.97 \times\) \(10^{15} \operatorname{rad} / \mathrm{s} ) t+\left(1.99 \times 10^{7} \mathrm{rad} / \mathrm{m}\right) x ] .\) (a) What are the amplitudes of the electric and magnetic fields of this wave? (b) What are the frequency, wavelength, and period of the wave? Is this light visible to humans? (c) What is the speed of the wave?

\(\bullet\) A sinusoidal electromagnetic wave having a magnetic field of amplitude 1.25 \(\mu\) and a wavelength of 432 nm is traveling in the \(+x\) direction through empty space. (a) What is the frequency of this wave? (b) What is the amplitude of the associated electric field? (c) Write the equations for the electric and magnetic fields as functions of \(x\) and \(t\) in the form of Equations \((23.3) .\)

\(\bullet\) Visible light. The wavelength of visible light ranges from 400 nm to 700 nm. Find the corresponding ranges of this light's (a) frequency, (b) angular frequency, (c) wave number.

\(\bullet\) Ultraviolet radiation. There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 \(\mathrm{nm}\) to 400 nm. It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between 280 \(\mathrm{nm}\) and \(320 \mathrm{nm},\) is much more dangerous, because it causes skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

\(\bullet\) Medical rays. Medical xays are taken with electromagnetic waves having a wavelength around 0.10 nm. What are the frequency, period, and wave number of such waves?

\(\bullet\) Radio station \(\mathrm{WCCO}\) in Minneapolis broadcasts at a frequency of 830 \(\mathrm{kHz}\) . At a point some distance from the transmitter, the magnetic-field amplitude of the electromagnetic wave from \(\mathrm{WCCO}\) is \(4.82 \times 10^{-11} \mathrm{T}\) . Calculate (a) the wavelength, (b) the wave number, (c) the angular frequency, and (d) the electric-field amplitude.

\(\bullet$$\bullet \mathrm{A}\) sinusoidal electromagnetic wave of frequency \(6.10 \times 10^{14} \mathrm{Hz}\) travels in vacuum in the \(+x\) -direction. The magnetic field is parallel to the \(y\) -axis and has amplitude \(5.80 \times 10^{-4} \mathrm{T} .\) (a) Find the magnitude and direction of the electric field. (b) Write the wave functions for the electric and magnetic fields in the form of Equations \((23.3).\)

\(\bullet\) Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) \(\vec{E}\) in the \(+x\) direction, \(\vec{B}\) in the \(+y\) direction. (b) \(\vec{E}\) in the \(-y\) direction, \(\vec{B}\) in the \(+x\) direction. (c) \(\vec{\boldsymbol{E}}\) in the \(+z\) direction, \(\vec{\boldsymbol{B}}\) in the \(-x\) direction. (d) \(\vec{\boldsymbol{E}}\) in the \(+y\) direction, \(\vec{\boldsymbol{B}}\) in the \(-z\) direction.

\(\bullet\) Laboratory lasers. He-Ne lasers are often used in physics demonstrations. They produce light of wavelength 633 \(\mathrm{nm}\) and a power of 0.500 \(\mathrm{mW}\) spread over a cylindrical beam 1.00 \(\mathrm{mm}\) in diameter (although these quantities can vary). (a) What is the intensity of this laser beam? (b) What are the maximum values of the electric and magnetic fields? (c) What is the average energy density in the laser beam?

\(\bullet\) Fields from a lightbulb. We can reasonably model a 75 \(\mathrm{W}\) incandescent lightbulb as a sphere 6.0 \(\mathrm{cm}\) in diameter. Typically, only about 5\(\%\) of the energy goes to visible light; the rest goes largely to nonvisible infrared radiation. (a) What is the visible light intensity (in \(\mathrm{W} / \mathrm{m}^{2} )\) at the surface of the bulb? (b) What are the amplitudes of the electric and magnetic fields at this surface, for a sinusoidal wave with this intensity?

\(\bullet\) Threshold of vision. Under controlled darkened conditions in the laboratory, a light receptor cell on the retina of a person's eye can detect a single photon (more on photons in Chapter 28 ) of light of wavelength 505 \(\mathrm{nm}\) and having an energy of \(3.94 \times 10^{-19} \mathrm{J} .\) We shall assume that this energy is absorbed by a single cell during one period of the wave. Cells of this kind are called rods and have a diameter of approximately 0.0020 \(\mathrm{mm} .\) What is the intensity (in \(\mathrm{W} / \mathrm{m}^{2} )\) delivered to a rod?

\(\bullet\) \(\cdot\) High-energy cancer treatment. Scientists are working on a new technique to kill cancer cells by zapping them with ultrahigh-energy (in the range of \(10^{12}\) W) pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk 5.0\(\mu \mathrm{m}\) in diameter, with the pulse lasting for 4.0 \(\mathrm{ns}\) with an average power of \(2.0 \times 10^{12} \mathrm{W}\) . We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in \(\mathrm{W} / \mathrm{m}^{2} )\) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?

\(\bullet\) At the floor of a room, the intensity of light from bright overhead lights is 8.00 \(\mathrm{W} / \mathrm{m}^{2} .\) Find the radiation pressure on a totally absorbing section of the floor.

\(\bullet\) The intensity at a certain distance from a bright light source is 6.00 \(\mathrm{W} / \mathrm{m}^{2} .\) Find the radiation pressure (in pascals and in atmospheres) on (a) a totally absorbing surface and (b) a totally reflecting surface.

\(\bullet$$\bullet\) A sinusoidal electromagnetic wave from a radio station passes perpendicularly through an open window that has area 0.500 \(\mathrm{m}^{2} .\) At the window, the electric field of the wave has rea value 0.0200 \(\mathrm{V} / \mathrm{m} .\) How much energy does this wave carry through the window during a 30.0 s commercial?

\(\bullet$$\bullet\) Two sources of sinusoidal electromagnetic waves have average powers of 75 \(\mathrm{W}\) and 150 \(\mathrm{W}\) and emit uniformly in all directions. At the same distance from each source, what is the ratio of the maximum electric field for the 150 \(\mathrm{W}\) source to that of the 75 \(\mathrm{W}\) source?

\(\bullet$$\bullet\) Radiation falling on a perfectly reflecting surface produces an average pressure \(p .\) If radiation of the same intensity falls on a perfectly absorbing surface and is spread over twice the area, what is the pressure at that surface in terms of \(p ?\)

\(\bullet$$\bullet\) A sinusoidal electromagnetic wave emitted by a cellular phone has a wavelength of 35.4 \(\mathrm{cm}\) and an electric field amplitude of \(5.40 \times 10^{-2} \mathrm{V} / \mathrm{m}\) at a distance of 250 \(\mathrm{m}\) from the antenna. Calculate: (a) the frequency of the wave; (b) the magnetic-field amplitude; (c) the intensity of the wave.

\(\bullet\) A light beam travels at \(1.94 \times 10^{8} \mathrm{m} / \mathrm{s}\) in quartz. The wavelength of the light in quartz is 355 \(\mathrm{nm}\) . (a) What is the index of refraction of quartz at this wavelength? (b) If this same light travels through air, what is its wavelength there?

\(\bullet$$\bullet\) Using a fast-pulsed laser and electronic timing circuitry, you find that light travels 2.50 \(\mathrm{m}\) within a plastic rod in 11.5 \(\mathrm{ns} .\) What is the refractive index of the plastic?

\(\bullet\) Light with a frequency of \(5.80 \times 10^{14}\) Hz travels in a block of glass that has an index of refraction of \(1.52 .\) What is the wavelength of the light (a) in vacuum and (b) in the glass?

\(\bullet\) The speed of light with a wavelength of 656 \(\mathrm{nm}\) in heavy flint glass is \(1.82 \times 10^{8} \mathrm{m} / \mathrm{s} .\) What is the index of refraction of the glass at this wavelength?

\(\bullet\) Light inside the eye. The vitreous humor, a transparent, gelatinous fluid that fills most of the eyeball, has an index of refraction of 1.34 . Visible light ranges in wavelength from 400 \(\mathrm{nm}\) (violet) to \(700 \mathrm{nm}(\mathrm{red}),\) as measured in air. This light travels through the vitreous humor and strikes the rods and cones at the surface of the retina. What are the ranges of (a) the wavelength, (b) the frequency, and (c) the speed of the light just as it approaches the retina within the vitreous humor?

\(\bullet$$\bullet\) A 1.55 -m-tall fisherman stands at the edge of a lake, being watched by a suspicious trout who is 3.50 \(\mathrm{m}\) from the fisherman in the horizontal direction and 45.0 \(\mathrm{cm}\) below the surface of the water. At what angle from the vertical does the fish see the top of the fisherman's head?

\(\bullet\) A glass plate having parallel faces and a refractive index of 1.58 lies at the bottom of a liquid of refractive index \(1.70 . \mathrm{A}\) ray of light in the liquid strikes the top of the glass at an angle of incidence of \(62.0^{\circ} .\) Compute the angle of refraction of this light in the glass.

\(\bullet\) A beam of light in air makes an angle of \(47.5^{\circ}\) with the surface (not the normal) of a glass plate having a refractive index of 1.66 (a) What is the angle between the reflected part of the beam and the surface of the glass? (b) What is the angle between the refracted beam and the surface (not the normal) of the glass?

\(\bullet$$\bullet\) You (height of your eyes above the water, 1.75 \(\mathrm{m}\) ) are standing 2.00 \(\mathrm{m}\) from the edge of a 2.50 -m-deep swimming pool. You notice that you can barely see your cell phone, which went missing a few minutes before, on the bottom of the pool. How far from the side of the pool is your cell phone?

\(\bullet$$\bullet\) A parallel-sided plate of glass having a refractive index of 1.60 is in contact with the surface of water in a tank. A ray coming from above makes an angle of incidence of \(32.0^{\circ}\) with the top surface of the glass. What angle does this ray make with the normal in the water?

\(\bullet\) A ray of light in diamond (index of refraction 2.42\()\) is incident on an interface with air. What is the largest angle the ray can make with the normal and not be totally reflected back into the diamond?

\(\bullet\) The critical angle for total internal reflection at a liquid-air interface is \(42.5^{\circ} .\) (a) If a ray of light traveling in the liquid has an angle of incidence of \(35.0^{\circ}\) at the interface, what angle does the refracted ray in the air make with the normal? (b) If a ray of light traveling in air has an angle of incidence of \(35.0^{\circ}\) at the interface, what angle does the refracted ray in the liquid make with the normal?

\(\bullet$$\bullet\) A ray of light is traveling in a glass cube that is totally immersed in water. You find that if the ray is incident on the glass-water interface at an angle to the normal greater than \(48.7^{\circ},\) no light is refracted into the water. What is the refractive index of the glass?

\(\bullet$$\bullet\) An optical fiber consists of an outer "cladding" layer and an inner core with a slightly higher index of refraction. Light rays entering the core are trapped inside by total internal reflection and forced to travel along the fiber (see Figure \(23.59 ) .\) Suppose the cladding has an index of refraction of 1.46 and the core has an index of refraction of \(1.48 .\) Calculate the largest angle \(\theta\) between a light ray and the longitudinal axis of the fiber ( see the figure) for which the ray will be totally internally reflected at the core/cladding boundary.

\(\bullet\) A beam of light strikes a sheet of glass at an angle of \(57.0^{\circ}\) with the normal in air. You observe that red light makes an angle of \(38.1^{\circ}\) with the normal in the glass, while violet light makes a \(36.7^{\circ}\) angle. (a) What are the indexes of refraction of this glass for these colors of light? (b) What are the speeds of red and violet light in the glass?

\(\bullet\) The indices of refraction for violet light \((\lambda=400 \mathrm{nm})\) and red light \((\lambda=700 \mathrm{nm})\) in diamond are 2.46 and \(2.41,\) respectively. A ray of light traveling through air strikes the diamond surface at an angle of \(53.5^{\circ}\) to the normal. Calculate the angular separation between these two colors of light in the refracted ray.

\(\bullet\) Unpolarized light with intensity \(I_{0}\) is incident on an ideal polarizing filter. The emerging light strikes a second ideal polarizing filter whose axis is at \(41.0^{\circ}\) to that of the first. Determine (a) the intensity of the beam after it has passed through the second polarizer and (b) its state of polarization.

\(\bullet\) Two ideal polarizing filters are oriented so that they transmit the maximum amount of light when unpolarized light is shone on them. To what fraction of its maximum value \(I_{0}\) is the intensity of the transmitted light reduced when the second filter is rotated through (a) \(22.5^{\circ},\) (b) \(45.0^{\circ},\) and (c) \(67.5^{\circ} ?\)

\(\bullet\) Light of original intensity \(I_{0}\) passes through two ideal polarizing filters having their polarizing axes oriented as shown in Figure \(23.62 .\) You want to adjust the angle \(\phi\) so that the intensity at point \(P\) is equal to \(I_{0} / 10 .\) (a) If the original light is unpolarized, what should \(\phi\) be? (b) If the original light is linearly polarized in the same direction as the polarizing axis of the first polarizer the light reaches, what should \(\phi\) be?

\(\bullet\) The polarizing angle for light in air incident on a glass plate is \(57.6^{\circ} .\) What is the index of refraction of the glass?

\(\bullet$$\bullet\) A beam of polarized light passes through a polarizing filter. When the angle between the polarizing axis of the filter and the direction of polarization of the light is \(\theta\) , the intensity of the emerging beam is \(I\) . If you instead want the intensity to be \(I / 2,\) what should be the angle (in terms of \(\theta )\) between the polarizing angle of the filter and the original direction of polarization of the light?

\(\bullet$$\bullet\) A beam of unpolarized light in air is incident at an angle of \(54.5^{\circ}\) (with respect to the normal) on a plane glass surface. The reflected beam is completely linearly polarized. (a) What is the refractive index of the glass? (b) What is the angle of refraction of the transmitted beam?

\(\bullet$$\bullet\) Plane-polarized light passes through two polarizers whose axes are oriented at \(35.0^{\circ}\) to each other. If the intensity of the original beam is reduced to \(15.0 \%,\) what was the polarization direction of the original beam, relative to the first polarizer?

\(\bullet$$\bullet\) A plane sinusoidal electromagnetic wave in air has a wave- length of 3.84 \(\mathrm{cm}\) and an \(\vec{\boldsymbol{E}}\) field amplitude of 1.35 \(\mathrm{V} / \mathrm{m}\) . (a) What is the frequency of the wave? (b) What is the \(\vec{\boldsymbol{B}}\) field amplitude? (c) What is the intensity? (d) What average force does this radiation exert perpendicular to its direction of propagation on a totally absorbing surface with area 0.240 \(\mathrm{m}^{2}\) ?

\(\bullet$$\bullet\) A powerful searchlight shines on a man. The man's cross- sectional area is 0.500 \(\mathrm{m}^{2}\) perpendicular to the light beam, and the intensity of the light at his location is 36.0 \(\mathrm{kW} / \mathrm{m}^{2}\) . He is wearing black clothing, so that the light incident on him is totally absorbed. What is the magnitude of the force the light beam exerts on the man? Do you think he could sense this force?

\(\bullet$$\bullet\) Laser surgery. Very short pulses of high-intensity laser beams are used to repair detached portions of the retina of the eye. The brief pulses of energy absorbed by the retina welds the detached portion back into place. In one such procedure, a laser beam has a wavelength of 810 \(\mathrm{nm}\) and delivers 250 \(\mathrm{mW}\) of power spread over a circular spot 510\(\mu \mathrm{m}\) in diameter. The vitreous humor (the transparent fluid that fills most of the eye) has an index of refraction of 1.34 . (a) If the laser pulses are each 1.50 \(\mathrm{ms}\) long, how much energy is delivered to the retina with each pulse? (b) What average pressure does the pulse of the laser beam exert on the retina as it is fully absorbed by the circular spot? (c) What are the wavelength and frequency of the laser light inside the vitreous humor of the eye? (d) What are the maximum values of the electric and magnetic fields in the laser beam?

\(\bullet$$\bullet\) A small helium-neon laser emits red visible light with a power of 3.20 \(\mathrm{mW}\) in a beam that has a diameter of 2.50 \(\mathrm{mm}\) . (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a 1.00 \(\mathrm{m}\) length of the beam?

\(\bullet$$\bullet\) A thick layer of oil is floating on the surface of water in a tank. A beam of light traveling in the oil is incident on the water interface at an angle of \(30.0^{\circ}\) from the normal. The refracted beam travels in the water at an angle of \(45.0^{\circ}\) from the normal. What is the refractive index of the oil?

\(\bullet$$\bullet\) A thin beam of light in air is incident on the surface of a lanthanum flint glass plate having a refractive index of 1.80 . What is the angle of incidence, \(\theta_{a}\) of the beam with this plate, for which the angle of refraction is \(\theta_{a} / 2 ?\) Both angles are measured relative to the normal.