Chapter 32

32.1 Legend says that Archimedes set the Roman fleet on fire as it was invading Syracuse. Archimedes created a huge ______ mirror, and he focused the Sun's rays on the Roman vessels. a) plane c) parabolic focusing b) parabolic diverging

Which of the following interface combinations has the smallest critical angle? a) light traveling from ice to diamond b) light traveling from quartz to lucite c) light traveling from diamond to glass d) light traveling from lucite to diamond e) light traveling from lucite to quartz

For specular reflection of a light ray, the angle of incidence a) must be equal to the angle of reflection. b) is always less than the angle of reflection. c) is always greater than the angle of reflection. d) is equal to \(90^{\circ}\) - the angle of reflection. e) may be greater than, less than, or equal to the angle of reflection.

Standing by a pool filled with water, under what condition will you see a reflection of the scenery on the opposite side through total internal reflection of the light from the scenery? a) Your eyes are level with the water. b) You observe the pool at an angle of \(41.8^{\circ}\) c) Under no condition. d) You observe the pool at an angle of \(48.2^{\circ}\)

You are using a mirror and a camera to make a self portrait. You focus the camera on yourself through the mirror. The mirror is a distance \(\mathrm{D}\) away from you. To what distance should you set the range of focus on the camera? a) \(D\) b) \(2 \mathrm{D}\) c) \(\mathrm{D} / 2\) d) \(4 \mathrm{D}\)

What is the magnification for a plane mirror? a) +1 c) greater than +1 b) -1 d) not defined for a plane mirror

If you look at an object at the bottom of a pool, the pool looks less deep than it actually is. a) From what you have learned, calculate how deep a pool seems to be if it is actually 4 feet deep and you look directly down on it. The refractive index of water is \(1.33 .\) b) Would the pool look more or less deep if you look at it from an angle other than vertical? Answer this qualitatively, without using an equation.

Why does refraction happen? That is, what is the physical reason a wave moves in a new medium with a different velocity than it did in the original medium?

Many fiber-optics devices have minimum specified bending angles. Why?

A physics student is eying a steel drum, the top part of which has the approximate shape of a concave spherical surface. The surface is sufficiently polished that she can just barely make out the reflection of her finger when she places it above the drum. As she slowly moves her finger toward the surface and then away from it, you ask her what she is doing. She replies that she is estimating the radius of curvature of the drum. How can she do that?

Answer as true or false with an explanation for the following: The wavelength of He-Ne laser light in water is less than its wavelength in the air. (The refractive index of water is \(1.33 .\)

Among the instruments Apollo astronauts left on the Moon were reflectors used to bounce laser beams back to Earth. These made it possible to measure the distance from the Earth to the Moon with unprecedented precision (uncertainties of a few centimeters out of \(384,000 \mathrm{~km}\) ), for the study both of celestial mechanics and of plate tectonics on Earth. The reflectors consisted not of ordinary mirrors, but of arrays of corner cubes, each consisting of three square plane mirrors fixed perpendicular to each other, as adjacent faces of a cube. Why? Explain the function and advantages of this design.

A \(45^{\circ}-45^{\circ}-90^{\circ}\) triangular prism can be used to reverse a light beam: The light enters perpendicular to the hypotenuse of the prism, reflects off each leg, and emerges perpendicular to the hypotenuse again. The surfaces of the prism are not silvered. If the prism is made of glass with in dex of refraction \(n_{\text {glass }}=1.520\) and the prism is surrounded by air, the light beam will be reflected with a minimum loss of intensity (there are reflection losses as the light enters and leaves the prism). a) Will this work if the prism is under water, which has index of refraction \(n_{\mathrm{H}_{2} \mathrm{O}}=1.333 ?\) b) Such prisms are used, in preference to mirrors, to bend the optical path in quality binoculars. Why?

You are under water in a pond and look up at the smooth surface of the water, noticing the sun in the sky. Is the sun in fact higher in the sky than it appears to you while under water, or is it lower?

A solar furnace uses a large parabolic mirror (mirrors several stories high have been constructed) to focus the light of the Sun to heat a target. A large solar furnace can melt metals. Is it possible to attain temperatures exceeding \(6000 \mathrm{~K}\) (the temperature of the photo sphere of the Sun) in a solar furnace? How, or why not?

A person sits \(1.0 \mathrm{~m}\) in front of a plane mirror. What is the location of the image?

Even the best mirrors absorb or transmit some of the light incident on them. The highest-quality mirrors might reflect \(99.997 \%\) of incident light intensity. Suppose a cubical "room, \(3.00 \mathrm{~m}\) on an edge, were constructed with such mirrors for the walls, floor, and ceiling. How slowly would such a room get dark? Estimate the time required for the intensity of light in such a room to fall to \(1.00 \%\) of its initial value after the only light source in the room is switched off.

The radius of curvature of a convex mirror is \(-25 \mathrm{~cm} .\) What is its focal length?

Convex mirrors are often used in side view mirrors on cars. Many such mirrors display the warning "Objects in mirror are closer than they appear." Assume a convex mirror has a radius of curvature of \(14.0 \mathrm{~m}\) and that there is a car that is \(11.0 \mathrm{~m}\) behind the mirror. For a flat mirror, the image distance would be \(11.0 \mathrm{~m}\) and the magnification would be 1\. Find the image distance and magnification for this mirror.

A \(5.00-\mathrm{cm}\) object is placed \(30.0 \mathrm{~cm}\) away from a convex mirror with a focal length of \(-10.0 \mathrm{~cm}\). Determine the size, orientation, and position of the image.

The magnification of a convex mirror is \(0.60 \times\) for an object \(2.0 \mathrm{~m}\) from the mirror. What is the focal length of this mirror?

An object is located at a distance of \(100 . \mathrm{cm}\) from a concave mirror of focal length \(20.0 \mathrm{~cm}\). Another concave mirror of focal length \(5.00 \mathrm{~cm}\) is located \(20.0 \mathrm{~cm}\) in front of the first concave mirror. The reflecting sides of the two mirrors face each other. What is the location of the final image formed by the two mirrors and the total magnification by the combination?

The shape of an elliptical mirror is described by the curve \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\) with semi major axis \(a\) and semi minor axis \(b\). The foci of this ellipse are at points \((c, 0)\) and \((-c, 0)\) with \(c=\left(a^{2}-b^{2}\right)^{1 / 2}\). Show that any light ray in the \(x y\) -plane, which passes through one focus, is reflected through the other. "Whispering galleries" make use of this phenomenon with sound waves.

What is the speed of light in crown glass, whose index of refraction is \(1.52 ?\)

An optical fiber with an index of refraction of 1.5 is used to transport light of wavelength \(400 \mathrm{nm}\). What is the critical angle for light to transport through this fiber without loss? If the fiber is immersed in water? In oil?

A light ray is incident from water of index of refraction 1.33 on a plate of glass whose index of refraction is 1.73. What is the angle of incidence, to have fully polarized reflected light?

Use Fermat's Principle to derive the law of reflection.

Fermat's Principle, from which geometric optics can be derived, states that light travels by a path that minimizes the time of travel between the points. Consider a light beam that travels a horizontal distance \(D\) and a vertical distance \(h\), through two large flat slabs of material, with a vertical interface between the materials. One material has a thickness \(D / 2\) and index of refraction \(n_{1},\) and the second material has a thickness \(D / 2\) and index of refraction \(n_{2} .\) Determine the equation involving the indices of refraction and angles from horizontal that the light makes at the interface \(\left(\theta_{1}\right.\) and \(\theta_{2}\) ) which minimize the time for this travel.

Suppose your height is \(2.0 \mathrm{~m}\) and you are standing \(50 \mathrm{~cm}\) in front of a plane mirror. a) What is the image distance? b) What is the image height? c) Is the image inverted or upright? d) Is the image real or virtual?

A light ray of wavelength 700 . \(\mathrm{nm}\) traveling in air \(\left(n_{1}=1.00\right)\) is incident on a boundary with a liquid \(\left(n_{2}=1.63\right) .\) a) What is the frequency of the refracted ray? b) What is the speed of the refracted ray? c) What is the wavelength of the refracted ray?

You have a spherical mirror with a radius of curvature of \(+20.0 \mathrm{~cm}\) (so it is concave facing you). You are looking at an object whose size you want to double in the image, so you can see it better. Where should you put the object? Where will the image be, and will it be real or virtual?

You are submerged in a swimming pool. What is the maximum angle at which you can see light coming from above the pool surface? That is, what is the angle for total internal reflection from water into air?

One of the factors that cause a diamond to sparkle is its relatively small critical angle. Compare the critical angle of diamond in air compared to that of diamond in water.

What kinds of images, virtual or real, are formed by a converging mirror when the object is placed a distance away from the mirror that is a) beyond the center of curvature of the mirror, b) between the center of curvature and half the center of curvature, and c) closer than half of the center of curvature.

Reflection and refraction, like all classical features of light and other electromagnetic waves, are governed by the Maxwell equations. The Maxwell equations are time-reversal invariant, which means that any solution of the equations reversed in time is also a solution. a) Suppose some configuration of electric charge density \(\rho,\) current density \(\vec{j},\) electric field \(\vec{E},\) and magnetic field \(\vec{B}\) is a solution of the Maxwell equations. What is the corresponding time-reversed solution? b) How, then, do "one-way mirrors" work?