Chapter 1: The Nature of LightContent check — University Physics 3, Chapter 1: The Nature of Light

Table 1.1 shows that ethanol and fresh water have very similar indices of refraction. By what percentage do the speeds of light in these liquids differ?

In Table 1.1, the solid with the next highest index of refraction after diamond is zircon. If the diamond in Example 1.3 were replaced with a piece of zircon, what would be the new angle of refraction?

At the surface between air and water, light rays can go from air to water and from water to air. For which ray is there no possibility of total internal reflection?

In the preceding example, how much distance inside the block of crown glass would the red and the violet rays have to progress before they are separated by 1.0 mm?

In Example 1.6, we had $n_1<n_2$. If $n_2$ were decreased such that $n_1>n_2$ and the speed of light in medium 2 is faster than in medium 1, what would happen to the length of $AA\text{′}$? What would happen to the wave front $A\text{′}B\text{′}$ and the direction of the refracted ray?

Although we did not specify the direction in Example 1.7, let’s say the polarizing filter was rotated clockwise by $71.6\text{°}$ to reduce the light intensity by $90.0\text{\%}$. What would be the intensity reduction if the polarizing filter were rotated counterclockwise by $71.6\text{°}$?

What happens at Brewster’s angle if the original incident light is already $100\text{\%}$ vertically polarized?

Under what conditions can light be modeled like a ray? Like a wave?

Does the fact that the light flash from lightning reaches you before its sound prove that the speed of light is extremely large or simply that it is greater than the speed of sound? Discuss how you could use this effect to get an estimate of the speed of light.

Using the law of reflection, explain how powder takes the shine off of a person’s nose. What is the name of the optical effect?

Will light change direction toward or away from the perpendicular when it goes from air to water? Water to glass? Glass to air?

Explain why a person’s legs appear very short when wading in a pool. Justify your explanation with a ray diagram showing the path of rays from the feet to the eye of an observer who is out of the water.

A ring with a colorless gemstone is dropped into water. The gemstone becomes invisible when submerged. Can it be a diamond? Explain.

How can you use total internal reflection to estimate the index of refraction of a medium?

A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated by white light.

Does Huygens’s principle apply to all types of waves?

Can a sound wave in air be polarized? Explain.

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

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

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?

What is the speed of light in air? In crown glass?

In what substance in Table 1.1 is the speed of light $2.290\times {10}^8\text{m/s?}$

Components of some computers communicate with each other through optical fibers having an index of refraction $n=1.55.$ What time in nanoseconds is required for a signal to travel 0.200 m through such a fiber?

How far does light travel underwater during a time interval of $1.50\times {10}^{\mathrm{-6}}\text{s}$?

Show that when light reflects from two mirrors that meet each other at a right angle, the outgoing ray is parallel to the incoming ray, as illustrated below.

A flat mirror is neither converging nor diverging. To prove this, consider two rays originating from the same point and diverging at an angle $\theta$ (see below). Show that after striking a plane mirror, the angle between their directions remains $\theta .$

Unless otherwise specified, for the following problems, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively.

A light beam in air is incident on the surface of a pond, making an angle of $20\text{°}$ with respect to the surface. What are the angles of reflection and refraction?

Unless otherwise specified, for the following problems, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively.

A pencil flashlight submerged in water sends a light beam toward the surface at an angle of incidence of $30\text{°}$. What is the angle of refraction in air?

Unless otherwise specified, for the following problems, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively.

The path of a light beam in air goes from an angle of incidence of $35\text{°}$ to an angle of refraction of $22\text{°}$ when it enters a rectangular block of plastic. What is the index of refraction of the plastic?

Unless otherwise specified, for the following problems, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively.

(a) Using information in the preceding problem, find the height of the instructor’s head above the water, noting that you will first have to calculate the angle of incidence. (b) Find the apparent depth of the diver’s head below water as seen by the instructor.

(a) At the end of Example 1.4, it was stated that the critical angle for light going from diamond to air is $24.4\text{°}.$ Verify this. (b) What is the critical angle for light going from zircon to air?

At what minimum angle will you get total internal reflection of light traveling in water and reflected from ice?

You can determine the index of refraction of a substance by determining its critical angle. (a) What is the index of refraction of a substance that has a critical angle of $68.4\text{°}$ when submerged in water? What is the substance, based on Table 1.1? (b) What would the critical angle be for this substance in air?

Light rays fall normally on the vertical surface of the glass prism $\left(n=1.50\right)$ shown below. (a) What is the largest value for $\varphi$ such that the ray is totally reflected at the slanted face? (b) Repeat the calculation of part (a) if the prism is immersed in water.

A beam of white light goes from air into water at an incident angle of $75.0\text{°}$. At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted?

(a) A narrow beam of light containing yellow (580 nm) and green (550 nm) wavelengths goes from polystyrene to air, striking the surface at a $30.0\text{°}$ incident angle. What is the angle between the colors when they emerge? (b) How far would they have to travel to be separated by 1.00 mm?

A ray of 610-nm light goes from air into fused quartz at an incident angle of $55.0\text{°}$. At what incident angle must 470 nm light enter flint glass to have the same angle of refraction?

A narrow beam of white light enters a prism made of crown glass at a $45.0\text{°}$ incident angle, as shown below. At what angles, ${\theta }_{\text{R}}$ and ${\theta }_{\text{V}}$, do the red (660 nm) and violet (410 nm) components of the light emerge from the prism?

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

Suppose that in the preceding problem the light incident on ${\text{P}}_1$ is unpolarized. At the determined value of $\theta$, what fraction of the incident light passes through the combination?

What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity $1.00{\text{kW/m}}^2$ to reduce the intensity to $10.0{\text{W/m}}^2$?

Show that if you have three polarizing filters, with the second at an angle of $45.0\text{°}$ to the first and the third at an angle of $90.0\text{°}$ to the first, the intensity of light passed by the first will be reduced to $25.0\text{\%}$ 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.)

In order to rotate the polarization axis of a beam of linearly polarized light by $90.0\text{°}$, a student places sheets ${\text{P}}_1$ and ${\text{P}}_2$ with their transmission axes at $45.0\text{°}$ and $90.0\text{°}$, respectively, to the beam’s axis of polarization. (a) What fraction of the incident light passes through ${\text{P}}_1$ and (b) through the combination? (c) Repeat your calculations for part (b) for transmission-axis angles of $30.0\text{°}$ and $90.0\text{°}$, respectively.

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

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

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?

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 $\left(\text{Δ}x\right),$ given that the incident angle is $40.0\text{°}$ and the glass is 1.00 cm thick.

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

(a) Light reflected at $62.5\text{°}$ 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?

Unreasonable results Suppose light travels from water to another substance, with an angle of incidence of $10.0\text{°}$ and an angle of refraction of $14.9\text{°}$. (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?

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

(a) On a day when the intensity of sunlight is $1.00{\text{kW/m}}^2$, 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 $20.0\text{°}$. Assuming the sunlight is unpolarized and the polarizers are $100\text{\%}$ efficient, what is the initial rate of heating of the water in $\text{°}\text{C/s}$, assuming it is $80.0\text{\%}$ 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.

Consider sunlight entering Earth’s atmosphere at sunrise and sunset—that is, at a $90.0\text{°}$ 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.

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

$n=$ $\frac{\text{sin}\frac{1}{2}(\alpha +\varphi )}{\text{sin}\frac{1}{2}\varphi }$

where $\varphi$ is the vertex angle of the prism and $\alpha$ is the angle through which the beam has been deviated. If $\alpha =37.0\text{°}$ and the base angles of the prism are each $50.0\text{°},$ what is n?

The light incident on polarizing sheet ${\text{P}}_1$ is linearly polarized at an angle of $30.0\text{°}$ with respect to the transmission axis of ${\text{P}}_1$. Sheet ${\text{P}}_2$ is placed so that its axis is parallel to the polarization axis of the incident light, that is, also at $30.0\text{°}$ with respect to ${\text{P}}_1$. (a) What fraction of the incident light passes through ${\text{P}}_1$? (b) What fraction of the incident light is passed by the combination? (c) By rotating ${\text{P}}_2$, a maximum in transmitted intensity is obtained. What is the ratio of this maximum intensity to the intensity of transmitted light when ${\text{P}}_2$ is at $30.0\text{°}$ with respect to ${\text{P}}_1$?