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1.7: Polarization

Polarizing sunglasses are familiar to most of us. They have a special ability to cut the glare of light reflected from water or glass (Figure). They have this ability because of a wave characteristic of light called polarization. What is polarization? How is it produced? What are some of its uses? The answers to these questions are related to the wave character of light.

Figure 1.These two photographs of a river show the effect of a polarizing filter in reducing glare in light reflected from the surface of water. Part (b) of this figure was taken with a polarizing filter and part (a) was not. As a result, the reflection of clouds and sky observed in part (a) is not observed in part (b). Polarizing sunglasses are particularly useful on snow and water. (credit a and credit b: modifications of work by “Amithshs”/Wikimedia Commons)

Malus’s Law

Light is one type of electromagnetic (EM) wave. As noted in the previous chapter on Electromagnetic Waves, EM waves are transverse waves consisting of varying electric and magnetic fields that oscillate perpendicular to the direction of propagation (Figure). However, in general, there are no specific directions for the oscillations of the electric and magnetic fields; they vibrate in any randomly oriented plane perpendicular to the direction of propagation. Polarization is the attribute that a wave’s oscillations do have a definite direction relative to the direction of propagation of the wave. (This is not the same type of polarization as that discussed for the separation of charges.) Waves having such a direction are said to be polarized. For an EM wave, we define the direction of polarization to be the direction parallel to the electric field. Thus, we can think of the electric field arrows as showing the direction of polarization, as in Figure.

Figure 2. An EM wave, such as light, is a transverse wave. The electric $$\overrightarrow{E}$$ and magnetic $$\overrightarrow{B}$$ fields are perpendicular to the direction of propagation. The direction of polarization of the wave is the direction of the electric field.

To examine this further, consider the transverse waves in the ropes shown in Figure. The oscillations in one rope are in a vertical plane and are said to be vertically polarized. Those in the other rope are in a horizontal plane and are horizontally polarized. If a vertical slit is placed on the first rope, the waves pass through. However, a vertical slit blocks the horizontally polarized waves. For EM waves, the direction of the electric field is analogous to the disturbances on the ropes.

Figure 3. The transverse oscillations in one rope (a) are in a vertical plane, and those in the other rope (b) are in a horizontal plane. The first is said to be vertically polarized, and the other is said to be horizontally polarized. Vertical slits pass vertically polarized waves and block horizontally polarized waves.

The Sun and many other light sources produce waves that have the electric fields in random directions (Figure(a)). Such light is said to be unpolarized, because it is composed of many waves with all possible directions of polarization. Polaroid materials—which were invented by the founder of the Polaroid Corporation, Edwin Land—act as a polarizing slit for light, allowing only polarization in one direction to pass through. Polarizing filters are composed of long molecules aligned in one direction. If we think of the molecules as many slits, analogous to those for the oscillating ropes, we can understand why only light with a specific polarization can get through. The axis of a polarizing filter is the direction along which the filter passes the electric field of an EM wave.

Figure 4. The slender arrow represents a ray of unpolarized light. The bold arrows represent the direction of polarization of the individual waves composing the ray. (a) If the light is unpolarized, the arrows point in all directions. (b) A polarizing filter has a polarization axis that acts as a slit passing through electric fields parallel to its direction. The direction of polarization of an EM wave is defined to be the direction of its electric field.

Figure shows the effect of two polarizing filters on originally unpolarized light. The first filter polarizes the light along its axis. When the axes of the first and second filters are aligned (parallel), then all of the polarized light passed by the first filter is also passed by the second filter. If the second polarizing filter is rotated, only the component of the light parallel to the second filter’s axis is passed. When the axes are perpendicular, no light is passed by the second filter.

Figure 5. The effect of rotating two polarizing filters, where the first polarizes the light. (a) All of the polarized light is passed by the second polarizing filter, because its axis is parallel to the first. (b) As the second filter is rotated, only part of the light is passed. (c) When the second filter is perpendicular to the first, no light is passed. (d) In this photograph, a polarizing filter is placed above two others. Its axis is perpendicular to the filter on the right (dark area) and parallel to the filter on the left (lighter area). (credit d: modification of work by P.P. Urone)

Only the component of the EM wave parallel to the axis of a filter is passed. Let us call the angle between the direction of polarization and the axis of a filter θ. If the electric field has an amplitude E, then the transmitted part of the wave has an amplitude Ecosθ (Figure). Since the intensity of a wave is proportional to its amplitude squared, the intensity I of the transmitted wave is related to the incident wave by

Note

$I=I_0cos^2θ$

where I0 is the intensity of the polarized wave before passing through the filter. This equation is known as Malus’s law.

Figure 6. A polarizing filter transmits only the component of the wave parallel to its axis, reducing the intensity of any light not polarized parallel to its axis.

Note

This Open Source Physics animation helps you visualize the electric field vectors as light encounters a polarizing filter. You can rotate the filter—note that the angle displayed is in radians. You can also rotate the animation for 3D visualization.

Example

Calculating Intensity Reduction by a Polarizing Filter

What angle is needed between the direction of polarized light and the axis of a polarizing filter to reduce its intensity by 90.0%?

Strategy

When the intensity is reduced by 90.0%, it is 10.0% or 0.100 times its original value. That is, I=0.100I0. Using this information, the equation I=I0cos2θ can be used to solve for the needed angle.

Solution

Solving the equation I=I0cos2θ for cosθ and substituting with the relationship between I and I0 gives

$cosθ=\frac{I}{I_0}=\frac{0.100I_0}{I_0}=0.3162.$

Solving for θ yields

$θ=cos^{−1}0.3162=71.6°.$

Significance

A fairly large angle between the direction of polarization and the filter axis is needed to reduce the intensity to 10.0% of its original value. This seems reasonable based on experimenting with polarizing films. It is interesting that at an angle of 45°, the intensity is reduced to 50% of its original value. Note that 71.6° is 18.4° from reducing the intensity to zero, and that at an angle of 18.4°, the intensity is reduced to 90.0% of its original value, giving evidence of symmetry.

Note

Exercise

Check Your Understanding Although we did not specify the direction in Example, let’s say the polarizing filter was rotated clockwise by 71.6° to reduce the light intensity by 90.0%. What would be the intensity reduction if the polarizing filter were rotated counterclockwise by 71.6°?

[Show Solution]

also 90.0%

Polarization by Reflection

By now, you can probably guess that polarizing sunglasses cut the glare in reflected light, because that light is polarized. You can check this for yourself by holding polarizing sunglasses in front of you and rotating them while looking at light reflected from water or glass. As you rotate the sunglasses, you will notice the light gets bright and dim, but not completely black. This implies the reflected light is partially polarized and cannot be completely blocked by a polarizing filter.

Figure illustrates what happens when unpolarized light is reflected from a surface. Vertically polarized light is preferentially refracted at the surface, so the reflected light is left more horizontally polarized. The reasons for this phenomenon are beyond the scope of this text, but a convenient mnemonic for remembering this is to imagine the polarization direction to be like an arrow. Vertical polarization is like an arrow perpendicular to the surface and is more likely to stick and not be reflected. Horizontal polarization is like an arrow bouncing on its side and is more likely to be reflected. Sunglasses with vertical axes thus block more reflected light than unpolarized light from other sources.

Figure 7. Polarization by reflection. Unpolarized light has equal amounts of vertical and horizontal polarization. After interaction with a surface, the vertical components are preferentially absorbed or refracted, leaving the reflected light more horizontally polarized. This is akin to arrows striking on their sides and bouncing off, whereas arrows striking on their tips go into the surface.

Since the part of the light that is not reflected is refracted, the amount of polarization depends on the indices of refraction of the media involved. It can be shown that reflected light is completely polarized at an angle of reflection θb given by

Note

$tanθ_b=\frac{n_2}{n_1}$

where n1 is the medium in which the incident and reflected light travel and n2 is the index of refraction of the medium that forms the interface that reflects the light. This equation is known as Brewster’s law and θb is known as Brewster’s angle, named after the nineteenth-century Scottish physicist who discovered them.

Note

This Open Source Physics animation shows incident, reflected, and refracted light as rays and EM waves. Try rotating the animation for 3D visualization and also change the angle of incidence. Near Brewster’s angle, the reflected light becomes highly polarized.

Example

Calculating Polarization by Reflection

(a) At what angle will light traveling in air be completely polarized horizontally when reflected from water? (b) From glass?

Strategy

All we need to solve these problems are the indices of refraction. Air has n1=1.00, water has n2=1.333, and crown glass has n′2=1.520. The equation $$tanθ_b=\frac{n_2}{n_1}$$ can be directly applied to find θb in each case.

Solution

a. Putting the known quantities into the equation

$tanθ_b=\frac{n_2}{n_1}$

gives

$tanθ_b=\frac{n_2}{n_1}=\frac{1.333}{1.00}=1.333.$

Solving for the angle θb yields

$θ_b=tan^{−1}1.333=53.1°.$

b. Similarly, for crown glass and air,

$tanθ′_b=\frac{n′_2}{n_1}=\frac{1.520}{1.00}=1.52.$

Thus,

$θ′_b=tan^{−1}1.52=56.7°.$

Significance

Light reflected at these angles could be completely blocked by a good polarizing filter held with its axis vertical. Brewster’s angle for water and air are similar to those for glass and air, so that sunglasses are equally effective for light reflected from either water or glass under similar circumstances. Light that is not reflected is refracted into these media. Therefore, at an incident angle equal to Brewster’s angle, the refracted light is slightly polarized vertically. It is not completely polarized vertically, because only a small fraction of the incident light is reflected, so a significant amount of horizontally polarized light is refracted.

Note

Exercise

Check Your Understanding What happens at Brewster’s angle if the original incident light is already 100% vertically polarized?

[Show Solution]

There will be only refraction but no reflection.

Atomic Explanation of Polarizing Filters

Polarizing filters have a polarization axis that acts as a slit. This slit passes EM waves (often visible light) that have an electric field parallel to the axis. This is accomplished with long molecules aligned perpendicular to the axis, as shown in Figure.

Figure 8. Long molecules are aligned perpendicular to the axis of a polarizing filter. In an EM wave, the component of the electric field perpendicular to these molecules passes through the filter, whereas the component parallel to the molecules is absorbed.

Figure illustrates how the component of the electric field parallel to the long molecules is absorbed. An EM wave is composed of oscillating electric and magnetic fields. The electric field is strong compared with the magnetic field and is more effective in exerting force on charges in the molecules. The most affected charged particles are the electrons, since electron masses are small. If an electron is forced to oscillate, it can absorb energy from the EM wave. This reduces the field in the wave and, hence, reduces its intensity. In long molecules, electrons can more easily oscillate parallel to the molecule than in the perpendicular direction. The electrons are bound to the molecule and are more restricted in their movement perpendicular to the molecule. Thus, the electrons can absorb EM waves that have a component of their electric field parallel to the molecule. The electrons are much less responsive to electric fields perpendicular to the molecule and allow these fields to pass. Thus, the axis of the polarizing filter is perpendicular to the length of the molecule.

Figure 9. Diagram of an electron in a long molecule oscillating parallel to the molecule. The oscillation of the electron absorbs energy and reduces the intensity of the component of the EM wave that is parallel to the molecule.

Polarization by Scattering

If you hold your polarizing sunglasses in front of you and rotate them while looking at blue sky, you will see the sky get bright and dim. This is a clear indication that light scattered by air is partially polarized. Figure helps illustrate how this happens. Since light is a transverse EM wave, it vibrates the electrons of air molecules perpendicular to the direction that it is traveling. The electrons then radiate like small antennae. Since they are oscillating perpendicular to the direction of the light ray, they produce EM radiation that is polarized perpendicular to the direction of the ray. When viewing the light along a line perpendicular to the original ray, as in the figure, there can be no polarization in the scattered light parallel to the original ray, because that would require the original ray to be a longitudinal wave. Along other directions, a component of the other polarization can be projected along the line of sight, and the scattered light is only partially polarized. Furthermore, multiple scattering can bring light to your eyes from other directions and can contain different polarizations.

Figure 10. Polarization by scattering. Unpolarized light scattering from air molecules shakes their electrons perpendicular to the direction of the original ray. The scattered light therefore has a polarization perpendicular to the original direction and none parallel to the original direction.

Photographs of the sky can be darkened by polarizing filters, a trick used by many photographers to make clouds brighter by contrast. Scattering from other particles, such as smoke or dust, can also polarize light. Detecting polarization in scattered EM waves can be a useful analytical tool in determining the scattering source.

A range of optical effects are used in sunglasses. Besides being polarizing, sunglasses may have colored pigments embedded in them, whereas others use either a nonreflective or reflective coating. A recent development is photochromic lenses, which darken in the sunlight and become clear indoors. Photochromic lenses are embedded with organic microcrystalline molecules that change their properties when exposed to UV in sunlight, but become clear in artificial lighting with no UV.

Liquid Crystals and Other Polarization Effects in Materials

Although you are undoubtedly aware of liquid crystal displays (LCDs) found in watches, calculators, computer screens, cellphones, flat screen televisions, and many other places, you may not be aware that they are based on polarization. Liquid crystals are so named because their molecules can be aligned even though they are in a liquid. Liquid crystals have the property that they can rotate the polarization of light passing through them by 90°. Furthermore, this property can be turned off by the application of a voltage, as illustrated in Figure. It is possible to manipulate this characteristic quickly and in small, well-defined regions to create the contrast patterns we see in so many LCD devices.

In flat screen LCD televisions, a large light is generated at the back of the TV. The light travels to the front screen through millions of tiny units called pixels (picture elements). One of these is shown in Figure(a) and (b). Each unit has three cells, with red, blue, or green filters, each controlled independently. When the voltage across a liquid crystal is switched off, the liquid crystal passes the light through the particular filter. We can vary the picture contrast by varying the strength of the voltage applied to the liquid crystal.

Figure 11. (a) Polarized light is rotated 90° by a liquid crystal and then passed by a polarizing filter that has its axis perpendicular to the direction of the original polarization. (b) When a voltage is applied to the liquid crystal, the polarized light is not rotated and is blocked by the filter, making the region dark in comparison with its surroundings. (c) LCDs can be made color specific, small, and fast enough to use in laptop computers and TVs.

Many crystals and solutions rotate the plane of polarization of light passing through them. Such substances are said to be optically active. Examples include sugar water, insulin, and collagen (Figure). In addition to depending on the type of substance, the amount and direction of rotation depend on several other factors. Among these is the concentration of the substance, the distance the light travels through it, and the wavelength of light. Optical activity is due to the asymmetrical shape of molecules in the substance, such as being helical. Measurements of the rotation of polarized light passing through substances can thus be used to measure concentrations, a standard technique for sugars. It can also give information on the shapes of molecules, such as proteins, and factors that affect their shapes, such as temperature and pH.

Figure 12. Optical activity is the ability of some substances to rotate the plane of polarization of light passing through them. The rotation is detected with a polarizing filter or analyzer.

Glass and plastic become optically active when stressed: the greater the stress, the greater the effect. Optical stress analysis on complicated shapes can be performed by making plastic models of them and observing them through crossed filters, as seen in Figure. It is apparent that the effect depends on wavelength as well as stress. The wavelength dependence is sometimes also used for artistic purposes.

Figure 13. Optical stress analysis of a plastic lens placed between crossed polarizers. (credit: “Infopro”/Wikimedia Commons)

Another interesting phenomenon associated with polarized light is the ability of some crystals to split an unpolarized beam of light into two polarized beams. This occurs because the crystal has one value for the index of refraction of polarized light but a different value for the index of refraction of light polarized in the perpendicular direction, so that each component has its own angle of refraction. Such crystals are said to be birefringent, and, when aligned properly, two perpendicularly polarized beams will emerge from the crystal (Figure). Birefringent crystals can be used to produce polarized beams from unpolarized light. Some birefringent materials preferentially absorb one of the polarizations. These materials are called dichroic and can produce polarization by this preferential absorption. This is fundamentally how polarizing filters and other polarizers work.

Figure 14. Birefringent materials, such as the common mineral calcite, split unpolarized beams of light into two with two different values of index of refraction.

Summary

• Polarization is the attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave. The direction of polarization is defined to be the direction parallel to the electric field of the EM wave.
• Unpolarized light is composed of many rays having random polarization directions.
• Unpolarized light can be polarized by passing it through a polarizing filter or other polarizing material. The process of polarizing light decreases its intensity by a factor of 2.
• The intensity, I, of polarized light after passing through a polarizing filter is I=I0cos2θ, where I0 is the incident intensity and θ is the angle between the direction of polarization and the axis of the filter.
• Polarization is also produced by reflection.
• Brewster’s law states that reflected light is completely polarized at the angle of reflection θb, known as Brewster’s angle.
• Polarization can also be produced by scattering.
• Several types of optically active substances rotate the direction of polarization of light passing through them.

Key Equations

 Speed of light $$c=2.99792458×10^8m/s≈3.00×10^8m/s$$ Index of refraction $$n=\frac{c}{v}$$ Law of reflection $$θ_r=θ_i$$ Law of refraction (Snell’s law) $$n_1sinθ_1=n_2sinθ_2$$ Critical angle $$θ_c=sin^{−1}(\frac{n_2}{n_1})~~for~~n_1>n_2$$ Malus’s law $$I=I_0cos^2θ$$ Brewster’s law $$tanθ_b=\frac{n_2}{n_1}$$

Conceptual Questions

Exercise

Can a sound wave in air be polarized? Explain.

[Show Solution]

No. Sound waves are not transverse waves.

Exercise

No light passes through two perfect polarizing filters with perpendicular axes. However, if a third polarizing filter is placed between the original two, some light can pass. Why is this? Under what circumstances does most of the light pass?

Exercise

Explain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.

[Show Solution]

Energy is absorbed into the filters.

Exercise

When particles scattering light are much smaller than its wavelength, the amount of scattering is proportional to 1/λ. Does this mean there is more scattering for small λ than large λ? How does this relate to the fact that the sky is blue?

Exercise

Using the information given in the preceding question, explain why sunsets are red.

[Show Solution]

Sunsets are viewed with light traveling straight from the Sun toward us. When blue light is scattered out of this path, the remaining red light dominates the overall appearance of the setting Sun.

Exercise

When light is reflected at Brewster’s angle from a smooth surface, it is 100% polarized parallel to the surface. Part of the light will be refracted into the surface. Describe how you would do an experiment to determine the polarization of the refracted light. What direction would you expect the polarization to have and would you expect it to be 100%?

Exercise

If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well. Why not?

[Show Solution]

The axis of polarization for the sunglasses has been rotated 90°.

Problems

Exercise

What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?

Exercise

The angle between the axes of two polarizing filters is 45.0°. By how much does the second filter reduce the intensity of the light coming through the first?

[Show Solution]

0.500

Exercise

Two polarizing sheets P1 and P2 are placed together with their transmission axes oriented at an angle θ to each other. What is θ when only 25% of the maximum transmitted light intensity passes through them?

Exercise

Suppose that in the preceding problem the light incident on P1 is unpolarized. At the determined value of θ, what fraction of the incident light passes through the combination?

[Show Solution]

0.125 or 1/8

Exercise

If you have completely polarized light of intensity 150W/m2, what will its intensity be after passing through a polarizing filter with its axis at an 89.0° angle to the light’s polarization direction?

Exercise

What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity 1.00kW/m2 to reduce the intensity to 10.0W/m2?

[Show Solution]

84.3°

Exercise

At the end of Example, it was stated that the intensity of polarized light is reduced to 90.0% of its original value by passing through a polarizing filter with its axis at an angle of 18.4° to the direction of polarization. Verify this statement.

Exercise

Show that if you have three polarizing filters, with the second at an angle of 45.0° to the first and the third at an angle of 90.0° to the first, the intensity of light passed by the first will be reduced to 25.0% of its value. (This is in contrast to having only the first and third, which reduces the intensity to zero, so that placing the second between them increases the intensity of the transmitted light.)

[Show Solution]

0.250 I0

Exercise

Three polarizing sheets are placed together such that the transmission axis of the second sheet is oriented at 25.0° to the axis of the first, whereas the transmission axis of the third sheet is oriented at 40.0° (in the same sense) to the axis of the first. What fraction of the intensity of an incident unpolarized beam is transmitted by the combination?

Exercise

In order to rotate the polarization axis of a beam of linearly polarized light by 90.0°, a student places sheets P1 and P2 with their transmission axes at 45.0° and 90.0°, respectively, to the beam’s axis of polarization. (a) What fraction of the incident light passes through P1 and (b) through the combination? (c) Repeat your calculations for part (b) for transmission-axis angles of 30.0° and 90.0°, respectively.

[Show Solution]

a. 0.500; b. 0.250; c. 0.187

Exercise

It is found that when light traveling in water falls on a plastic block, Brewster’s angle is 50.0°. What is the refractive index of the plastic?

Exercise

At what angle will light reflected from diamond be completely polarized?

[Show Solution]

67.54°

Exercise

What is Brewster’s angle for light traveling in water that is reflected from crown glass?

Exercise

A scuba diver sees light reflected from the water’s surface. At what angle will this light be completely polarized?

[Show Solution]

53.1°

Exercise

From his measurements, Roemer estimated that it took 22 min for light to travel a distance equal to the diameter of Earth’s orbit around the Sun. (a) Use this estimate along with the known diameter of Earth’s orbit to obtain a rough value of the speed of light. (b) Light actually takes 16.5 min to travel this distance. Use this time to calculate the speed of light.

Exercise

Cornu performed Fizeau’s measurement of the speed of light using a wheel of diameter 4.00 cm that contained 180 teeth. The distance from the wheel to the mirror was 22.9 km. Assuming he measured the speed of light accurately, what was the angular velocity of the wheel?

[Show Solution]

Exercise

Suppose you have an unknown clear substance immersed in water, and you wish to identify it by finding its index of refraction. You arrange to have a beam of light enter it at an angle of 45.0°, and you observe the angle of refraction to be 40.3°. What is the index of refraction of the substance and its likely identity?

Exercise

Shown below is a ray of light going from air through crown glass into water, such as going into a fish tank. Calculate the amount the ray is displaced by the glass (Δx), given that the incident angle is 40.0° and the glass is 1.00 cm thick.

[Show Solution]

3.72 mm

Exercise

Considering the previous problem, show that θ3 is the same as it would be if the second medium were not present.

Exercise

At what angle is light inside crown glass completely polarized when reflected from water, as in a fish tank?

[Show Solution]

41.2°

Exercise

Light reflected at 55.6° from a window is completely polarized. What is the window’s index of refraction and the likely substance of which it is made?

Exercise

(a) Light reflected at 62.5° from a gemstone in a ring is completely polarized. Can the gem be a diamond? (b) At what angle would the light be completely polarized if the gem was in water?

[Show Solution]

a. 1.92. The gem is not a diamond (it is zircon). b. 55.2°

Exercise

If θb is Brewster’s angle for light reflected from the top of an interface between two substances, and θ′b is Brewster’s angle for light reflected from below, prove that θb+θ′b=90.0°.

Exercise

Unreasonable results Suppose light travels from water to another substance, with an angle of incidence of 10.0° and an angle of refraction of 14.9°. (a) What is the index of refraction of the other substance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

[Show Solution]

a. 0.898; b. We cannot have n<1.00, since this would imply a speed greater than c. c. The refracted angle is too big relative to the angle of incidence.

Exercise

Unreasonable results Light traveling from water to a gemstone strikes the surface at an angle of 80.0° and has an angle of refraction of 15.2°. (a) What is the speed of light in the gemstone? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Exercise

If a polarizing filter reduces the intensity of polarized light to 50.0% of its original value, by how much are the electric and magnetic fields reduced?

[Show Solution]

0.707B1

Exercise

Suppose you put on two pairs of polarizing sunglasses with their axes at an angle of 15.0°. How much longer will it take the light to deposit a given amount of energy in your eye compared with a single pair of sunglasses? Assume the lenses are clear except for their polarizing characteristics.

Exercise

(a) On a day when the intensity of sunlight is 1.00kW/m2, a circular lens 0.200 m in diameter focuses light onto water in a black beaker. Two polarizing sheets of plastic are placed in front of the lens with their axes at an angle of 20.0°. Assuming the sunlight is unpolarized and the polarizers are 100% efficient, what is the initial rate of heating of the water in °C/s, assuming it is 80.0% absorbed? The aluminum beaker has a mass of 30.0 grams and contains 250 grams of water. (b) Do the polarizing filters get hot? Explain.

[Show Solution]

a. 1.69×10−2°C/s; b. yes

Challenge Problems

Exercise

Light shows staged with lasers use moving mirrors to swing beams and create colorful effects. Show that a light ray reflected from a mirror changes direction by 2θ when the mirror is rotated by an angle θ.

Exercise

Consider sunlight entering Earth’s atmosphere at sunrise and sunset—that is, at a 90.0° incident angle. Taking the boundary between nearly empty space and the atmosphere to be sudden, calculate the angle of refraction for sunlight. This lengthens the time the Sun appears to be above the horizon, both at sunrise and sunset. Now construct a problem in which you determine the angle of refraction for different models of the atmosphere, such as various layers of varying density. Your instructor may wish to guide you on the level of complexity to consider and on how the index of refraction varies with air density.

[Show Solution]

First part: 88.6°. The remainder depends on the complexity of the solution the reader constructs.

Exercise

A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown below. Show that if the fiber is made from crown glass, any incident ray will be totally internally reflected.

Exercise

A light ray falls on the left face of a prism (see below) at the angle of incidence θ for which the emerging beam has an angle of refraction θ at the right face. Show that the index of refraction n of the glass prism is given by

$n=\frac{sin\frac{1}{2}(α+ϕ)}{sin\frac{1}{2}ϕ}$

where ϕ is the vertex angle of the prism and α is the angle through which the beam has been deviated. If α=37.0° and the base angles of the prism are each 50.0°, what is n?

[Show Solution]

proof; 1.33

Exercise

If the apex angle ϕ in the previous problem is 20.0° and n=1.50, what is the value of α?

Exercise

The light incident on polarizing sheet P1 is linearly polarized at an angle of 30.0° with respect to the transmission axis of P1. Sheet P2 is placed so that its axis is parallel to the polarization axis of the incident light, that is, also at 30.0° with respect to P1. (a) What fraction of the incident light passes through P1? (b) What fraction of the incident light is passed by the combination? (c) By rotating P2, a maximum in transmitted intensity is obtained. What is the ratio of this maximum intensity to the intensity of transmitted light when P2 is at 30.0° with respect to P1?

[Show Solution]

a. 0.750; b. 0.563; c. 1.33

Exercise

Prove that if I is the intensity of light transmitted by two polarizing filters with axes at an angle θ and I′ is the intensity when the axes are at an angle 90.0°−θ, then I+I′=I0, the original intensity. (Hint: Use the trigonometric identities cos90.0°−θ=sinθ and cos2θ+sin2θ=1.)

Glossary

birefringent
refers to crystals that split an unpolarized beam of light into two beams
Brewster’s angle
angle of incidence at which the reflected light is completely polarized
Brewster’s law
tanθb=n2n1, where n1 is the medium in which the incident and reflected light travel and n2 is the index of refraction of the medium that forms the interface that reflects the light
direction of polarization
direction parallel to the electric field for EM waves
horizontally polarized
oscillations are in a horizontal plane
Malus’s law
where I0 is the intensity of the polarized wave before passing through the filter
optically active
substances that rotate the plane of polarization of light passing through them
polarization
attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave
polarized
refers to waves having the electric and magnetic field oscillations in a definite direction
unpolarized
refers to waves that are randomly polarized

vertically polarized
oscillations are in a vertical plane