Optics

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, the intensity of the emerging beam is I. If you now want the intensity to be I/2, what should be the angle (in terms of) between the polarizing angle of the filter and the original direction of polarization of the light?

Three polarizing filters are stacked, with the polarizing axis of the second and third filters at 23.0° and 62.0°, respectively, to that of the first. If unpolarized light is incident on the stack, the light has intensity 55.0 W/${\mathbf{cm}}^{\mathbf{2}}$ after it passes through the stack. If the incident intensity is kept constant but the second polarizer is removed, what is the intensity of the light after it has passed through the stack?

Unpolarized light of intensity 20.0 W/${\mathbf{cm}}^{\mathbf{2}}$ is incident on two polarizing filters. The axis of the first filter is at an angle of 25.0° counterclockwise from the vertical (viewed in the direction the light is traveling), and the axis of the second filter is at 62.0° counterclockwise from the vertical. What is the intensity of the light after it has passed through the second polarizer?

Three polarizing filters are stacked with the polarizing axes of the second and third at 45.0° and 90.0°, respectively, with that of the first. (a) If unpolarized light of intensity ${\mathit{I}}_{\mathbf{0}}$ is incident on the stack, find the intensity and state of polarization of light emerging from each filter. (b) If the second filter is removed, what is the intensity of the light emerging from each remaining filter?

A beam of white light passes through a uniform thickness of air. If the intensity of the scattered light in the middle of the green part of the visible spectrum is I, find the intensity (in terms of I) of scattered light in the middle of (a) the red part of the spectrum and (b) the violet part of the spectrum. Consult Table 32.1

A light beam is directed parallel to the axis of a hollow cylindrical tube. When the tube contains only air, the light takes 8.72 ns to travel the length of the tube, but when the tube is filled with a transparent jelly, the light takes 1.82 ns longer to travel its length. What is the refractive index of this jelly?

Physicians use high-frequency 1f = 195 MHz2 sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is 1480 m>s in muscle and 344 m/s in air. We define the index of refraction of a material for sound waves to be the ratio of the speed of sound in air to the speed of sound in the material. Snell’s law then applies to the refraction of sound waves. (a) At what angle from the normal does an ultrasound beam enter the heart if it leaves the lungs at an angle of 9.73° from the normal to the heart wall? (Assume that the speed of sound in the lungs is 344 m/s.) (b) What is the critical angle for sound waves in air incident on muscle?

In a physics lab, light with wavelength  travels in air from a laser to a photocell in  When a slab of glass  thick is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light  to travel from the laser to the photocell. What is the wavelength of the light in the glass?

A ray of light is incident in air on a block of a transparent solid whose index of refraction is n. If , what is the largest angle of incidence for which total internal reflection will occur at the vertical face (point A shown in Fig. P33.39)?

A light ray in air strikes the right angle prism shown in Fig. P33.40. The prism angle at B is 30.0° . This ray consists of two different wavelengths. When it emerges at face AB, it has been split into two different rays that diverge from each other by. Find the index of refraction of the prism for each of the two wavelengths

A ray of light travelling in a block of glass  is incident on the top surface at an angle with respect to the normal in the glass. If a layer of oil is placed on the top surface of the glass, the ray is totally reflected. What is the maximum possible index of refraction of the oil?

A ray of light travelling in air is incident at angle ua on one face of a prism made of glass. Part of the light refracts into the prism and strikes the opposite face at point A (Fig. P33.42). If the ray at A is at the critical angle, what is the value of?

A glass plate   thick, with an index of refraction of, is placed between a point source of light with wavelength  (in vacuum) and a screen. The distance from source to screen is . How many wavelengths are there between the source and the screen?

After a long day of driving you take a late-night swim in a motel swimming pool. When you go to your room, you realise that you have lost your room key in the pool. You borrow a powerful flashlight and walk around the pool, shining the light into it. The light shines on the key, which is lying on the bottom of the pool, when the flashlight is held  above the water surface and is directed at the surface a horizontal distance of  from the edge (Fig. P33.44). If the water here is  deep, how far is the key from the edge of the pool?

You sight along the rim of a glass with vertical sides so that the top rim is lined up with the opposite edge of the bottom (Fig. P33.45a). The glass is  =thin-walled, hollow cylinder  high. The diameter of the top and bottom of the glass is  While you keep your eye in the same position, a friend fills the glass with a transparent liquid, and you then see a dime that is lying at the center of the bottom of the glass (Fig. P33.45b). What is the index of refraction of the liquid?

Optical fibers are constructed with a cylindrical core surrounded by a sheath of cladding material. Common materials used are pure silica  for the cladding and silica doped with germanium for the core. (a) What is the critical angle  for light traveling in the core and reflecting at the interface with the cladding material? (b) The numerical aperture (NA) is defined as the angle of incidence  at the flat end of the cable for which light is incident on the core–cladding interface at angle ucrit (Fig. P33.46). Show that (c) What is the value of ui for ?

Answer

A thin layer of ice   floats on the surface of water  in a bucket. A ray of light from the bottom of the bucket travels upward through the water. (a) What is the largest angle with respect to the normal that the ray can make at the ice–water interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

A -45°-45°-90° prism is immersed in water. A ray of light is incident normally on one of its shorter faces. What is the minimum index of refraction that the prism must have if this ray is to be totally reflected within the glass at the long face of the prism?

The prism shown in Fig. has a refractive index of 1.66, and the angles A are . Two light rays m and n are parallel as they enter the prism. What is the angle between them after they emerge?

Light is incident normally on the short face of a 30°-60°-90° prism (Fig. P33.50). A drop of liquid is placed on the hypotenuse of the prism. If the index of refraction of the prism is 1.56, find the maximum index that the liquid may have for the light to be totally reflected.

When the sun is either rising or setting and appears to be just on the horizon, it is in fact below the horizon. The explanation for this seeming paradox is that light from the sun bends slightly when entering the earth’s atmosphere, as shown in Fig. Since our perception is based on the idea that light travels in straight lines, we perceive the light to be coming from an apparent position that is an angleabove the sun’s true position.

(a) Make the simplifying assumptions that the atmosphere has uniform density, and hence uniform index of refraction n, and extends to a height h above the earth’s surface, at which point it abruptly stops. Show that the angle  is given by

$\delta =arc\sin \left(\frac{nR}{R+h}\right)-arc\sin \left(\frac{R}{R+h}\right)$

where R = 6378 km is the radius of the earth. 

(b) Calculate $\delta$ using n = 1.0003 and h = 20 km. How does this compare to the angular radius of the sun, which is about one-quarter of a degree? (In actuality a light ray from the sun bends gradually, not abruptly, since the density and refractive index of the atmosphere change gradually with altitude.)

A horizontal cylindrical tank 2.20 m in diameter is half full of water. The space above the water is filled with a pressurized gas of unknown refractive index. A small laser can move along the curved bottom of the water and aims a light beam toward the center of the water surface (Fig.). You observe that when the laser has moved a distance S = 1.09 m or more (measured along the curved surface) from the lowest point in the water, no light enters the gas. 

(a) What is the index of refraction of the gas? 

(b) What minimum time does it take the light beam to travel from the laser to the rim of the tank when (i) S>1.09 m and (ii) S <1.09  m?

The angle of Deviation. The incident angle ua shown in Fig. is chosen so that the light passes symmetrically through the prism, which has refractive index n and apex angle A.

(a) Show that the angle of deviation d (the angle between the initial and final directions of the ray) is given by

$sin\frac{A+\delta }{2}=n\sin \frac{A}{2}$ 

(When the light passes through symmetrically, as shown, the angle of deviation is a minimum.) 

(b) Use the result of part (a) to find the angle of deviation for a ray of light passing symmetrically through a prism having three equal angles 1A = 60.0_2and n = 1.52. 

(c) A certain glass has a refractive index of 1.61 for red light (700 nm) and 1.66 for violet light (400 nm). If both colors pass through symmetrically, as described in part (a), and if A = 60.0_, find the difference between the angles of deviation for the two colors.

Light is incident in the air at an angle ua (Fig.) on the upper surface of a transparent plate, the surfaces of the plate being plane and parallel to each other. 

(a) Prove that ua = u_a. 

(b) Show that this is true for any number of different parallel

plates. 

(c) Prove that the lateral displacement d of the emergent beam is given by the relationship

$d=\frac{t\sin \left({\theta }_a-\theta {\text{'}}_b\right)}{\cos \theta {\text{'}}_b}$ where t is the thickness of the plate. (d) A ray of light is incident at an angle of 66.0_ on one surface of a glass plate 2.40 cm thick with an index of refraction of 1.80. The medium on either side of the plate is air. Find the lateral displacement between the incident and emergent rays.

A beam of unpolarized sunlight strikes the vertical plastic wall of a water tank at an unknown angle. Some of the light reflects from the wall and enters the water (Fig. P33.55). The refractive index of the plastic wall is 1.61. If the light that has been reflected from the wall into the water is observed to be completely polarized, what angle does this beam make with the normal inside the water?

In physics lab, you are studying the properties of four transparent liquids. You shine a ray of light (in air) onto the surface of each liquid—A, B, C, and D—one at a time, at a 60.0_ angle of incidence; you then measure the angle of refraction. The table gives your data: Liquid A B C D Ua 1 _2 36.4 40.5 32.1 35.2 The wavelength of the light when it is traveling in air is 589 nm.

(a) Find the refractive index of each liquid at this wavelength. Use Table 33.1 to identify each liquid, assuming that all four are listed in the table. 

(b) For each liquid, what is the dielectric constant K at the frequency of the 589-nm light? For each liquid, the relative permeability (Km) is very close to unity. (c) What is the frequency of the light in air and in each liquid?

Given small samples of three liquids, you are asked to determine their refractive indexes. However, you do not have enough of each liquid to measure the angle of refraction for light refracting from air into the liquid. Instead, for each liquid, you take a rectangular block of glass 1n = 1.522 and place a drop of the liquid on the top surface of the block. You shine a laser beam with wavelength 638 nm in vacuum at one side of the block and measure the largest angle of incidence ua for which there is total internal reflection at the interface between the glass and the liquid (Fig. P33.58). Your results are given in the table: Liquid A B C

Ua 1 _2 52.0 44.3 36.3 What is the refractive index of each liquid at this wavelength?

A beam of light traveling horizontally is made of an unpolarized component with intensity I0 and a polarized component with intensity Ip. The plane of polarization of the polarized component is oriented at an angle u with respect to the vertical. Figure P33.59 is a graph of the total intensity Itotal after the light passes through a polarizer versus the angle a that the polarizer’s axis makes with respect to the vertical. (a) What is the orientation of the polarized component? (That is, what is u?) (b) What are the values of I0 and Ip?

First, light with a plane of polarization at 45° to the horizontal shines on the insect. Which statement is true about the two types of cells?

(a) Both types detect this light. 

(b) Neither type detects this light. 

(c) Only type H detects the light. 

(d) Only type V detects the light.

Next, unpolarised light is reflected off a smooth horizontal piece of glass, and the reflected light shines on the insect. Which statement is true about the two types of cells? (a) When the light is directly above the glass, only type V detects the reflected light. (b) When the light is directly above the glass, only type H detects the reflected light. (c) When the light is about 35° above the horizontal, type V responds much more strongly than type H does. (d) When the light is about 35° above the horizontal, type H responds much more strongly than type V does.

To vary the angle as well as the intensity of polarized light, ordinary unpolarized light is passed through one polarizer with its transmission axis vertical, and then a second polarizer is placed between the first polarizer and the insect. When the light leaving the second polarizer has half the intensity of the original unpolarized light, which statement is true about the two types of cells? (a) Only type H detects this light. (b) Only type V detects this light. (c) Both types detect this light, but type H detects lighter. (d) Both types detect this light, but type V detects more light

A spherical mirror is cut in half horizontally. Will an image be formed by the bottom half of the mirror? If so, where will the image be formed?

A candle $\mathbf{4}\mathbf{.}\mathbf{85}\mathbf{ }\mathbf{cm}$ tall is $\mathbf{39}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{cm}$ to the left of a plane mirror. Where is the image formed by the mirror, and what is the height of this image?

For the situation shown in Fig. 34.3, is the image distance s′ positive or negative? Is the image real or virtual? Explain your answers

The image of a tree just covers the length of a plane mirror tall when the mirror is held 35cm from the eye. The tree is 28m from the mirror. What is its height?

The laws of optics also apply to electromagnetic waves invisible to the eye. A satellite TV dish is used to detect radio waves coming from orbiting satellites. Why is a curved reflecting surface (a “dish”) used? The dish is always concave, never convex; why? The actual radio receiver is placed on an arm and suspended in front of the dish. How far in front of the dish should it be placed?

A pencil that is 9cm long is held perpendicular to the surface of a plane mirror with the tip of the pencil lead 12cm from the mirror surface and the end of the eraser 21cm from the mirror surface. What is the length of the image of the pencil that is formed by the mirror? Which end of the image is closer to the mirror surface: the tip of the lead or the end of the eraser?

Explain why the focal length of a plane mirror is infinite, and explain what it means for the focal point to be at infinity.

A concave mirror has a radius of curvature of 34.0cm. (a) What is its focal length? (b) If the mirror is immersed in water (refractive index 1.33), what is its focal length?

If a spherical mirror is immersed in water, does its focal length change? Explain

For what range of object positions does a concave spherical mirror form a real image? What about a convex spherical mirror?

When a room has mirrors on two opposite walls, an infinite series of reflections can be seen. Discuss this phenomenon in terms of images. Why do the distant images appear fainter?

For a spherical mirror, if s = f , then s' = $\infty$, and the lateral magnification m is infinite. Does this make sense? If so, what does it mean?

You may have noticed a small convex mirror next to your bank’s ATM. Why is this mirror convex, as opposed to flat or concave? What considerations determine its radius of curvature?

A student claims that she can start a fire on a sunny day using just the sun’s rays and a concave mirror. How is this done? Is the concept of image relevant? Can she do the same thing with a convex mirror? Explain.

You hold a spherical salad bowl 60cm in front of your face with the bottom of the bowl facing you. The bowl is made of polished metal with a 35-cm radius of curvature. (a) Where is the image of your 5.0-cm tall nose located? (b) What are the image's size, orientation, and nature (real or virtual)?

A person looks at his reflection in the concave side of a shiny spoon. Is it right side up or inverted? Does it matter how far his face is from the spoon? What if he looks in the convex side?

There appears to be anambiguity for the case s = 10cm to whether s′ is $\mathbf{s}\mathbf{\text{'}}\mathbf{=}\mathbf{+}\mathbf{\infty }$and $\mathbf{-}\mathbf{\infty }$ whether the image is erect or inverted. How is this resolved? Or is it?

The bottom of the passenger-side mirror on your carnotes, “Objects in mirror are closer than they appear.” Is this true?Why?

How could you very quickly make an approximate measurement of the focal length of a converging lens? Could the samemethod be applied if you wished to use a diverging lens? Explain.