### Problems

## 1.1 The Propagation of Light

What is the speed of light in water? In glycerine?

Calculate the index of refraction for a medium in which the speed of light is $2.012\phantom{\rule{0.2em}{0ex}}\xc3\u2014\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s},$ and identify the most likely substance based on Table 1.1.

There was a major collision of an asteroid with the Moon in medieval times. It was described by monks at Canterbury Cathedral in England as a red glow on and around the Moon. How long after the asteroid hit the Moon, which is $3.84\phantom{\rule{0.2em}{0ex}}\xc3\u2014\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{km}$ away, would the light first arrive on Earth?

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?

Compare the time it takes for light to travel 1000 m on the surface of Earth and in outer space.

How far does light travel underwater during a time interval of $1.50\phantom{\rule{0.2em}{0ex}}\xc3\u2014\phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{\xe2\u02c6\u20196}}\phantom{\rule{0.2em}{0ex}}\text{s}$?

## 1.2 The Law of Reflection

Suppose a man stands in front of a mirror as shown below. His eyes are 1.65 m above the floor and the top of his head is 0.13 m higher. Find the height above the floor of the top and bottom of the smallest mirror in which he can see both the top of his head and his feet. How is this distance related to the manâ€™s height?

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.

On the Moonâ€™s surface, lunar astronauts placed a corner reflector, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round-trip time. What percent correction is needed to account for the delay in time due to the slowing of light in Earthâ€™s atmosphere? Assume the distance to the Moon is precisely $3.84\phantom{\rule{0.2em}{0ex}}\xc3\u2014\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m}$ and Earthâ€™s atmosphere (which varies in density with altitude) is equivalent to a layer 30.0 km thick with a constant index of refraction $n=1.000293.$

A flat mirror is neither converging nor diverging. To prove this, consider two rays originating from the same point and diverging at an angle $\mathrm{\xce\xb8}$ (see below). Show that after striking a plane mirror, the angle between their directions remains $\mathrm{\xce\xb8}.$

## 1.3 Refraction

Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively.

A light beam in air has an angle of incidence of $35\text{\xc2\xb0}$ at the surface of a glass plate. What are the angles of reflection and refraction?

A light beam in air is incident on the surface of a pond, making an angle of $20\text{\xc2\xb0}$ with respect to the surface. What are the angles of reflection and refraction?

When a light ray crosses from water into glass, it emerges at an angle of $30\text{\xc2\xb0}$ with respect to the normal of the interface. What is its angle of incidence?

A pencil flashlight submerged in water sends a light beam toward the surface at an angle of incidence of $30\text{\xc2\xb0}$. What is the angle of refraction in air?

Light rays from the Sun make a $30\text{\xc2\xb0}$ angle to the vertical when seen from below the surface of a body of water. At what angle above the horizon is the Sun?

The path of a light beam in air goes from an angle of incidence of $35\text{\xc2\xb0}$ to an angle of refraction of $22\text{\xc2\xb0}$ when it enters a rectangular block of plastic. What is the index of refraction of the plastic?

A scuba diver training in a pool looks at their instructor as shown below. What angle does the ray from the instructorâ€™s face make with the perpendicular to the water at the point where the ray enters? The angle between the ray in the water and the perpendicular to the water is $25.0\text{\xc2\xb0}$.

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

## 1.4 Total Internal Reflection

Verify that the critical angle for light going from water to air is $48.6\text{\xc2\xb0}$, as discussed at the end of Example 1.4, regarding the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air.

(a) At the end of Example 1.4, it was stated that the critical angle for light going from diamond to air is $24.4\text{\xc2\xb0}.$ Verify this. (b) What is the critical angle for light going from zircon to air?

An optical fiber uses flint glass clad with crown glass. What is the critical angle?

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

Suppose you are using total internal reflection to make an efficient corner reflector. If there is air outside and the incident angle is $45.0\text{\xc2\xb0}$, what must be the minimum index of refraction of the material from which the reflector is made?

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 $68.4\text{\xc2\xb0}$ 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?

A ray of light, emitted beneath the surface of an unknown liquid with air above it, undergoes total internal reflection as shown below. What is the index of refraction for the liquid and its likely identification?

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 $\mathrm{\xcf\u2022}$ 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.

## 1.5 Dispersion

(a) What is the ratio of the speed of red light to violet light in diamond, based on Table 1.2? (b) What is this ratio in polystyrene? (c) Which is more dispersive?

A beam of white light goes from air into water at an incident angle of $75.0\text{\xc2\xb0}$. At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted?

By how much do the critical angles for red (660 nm) and violet (410 nm) light differ in a diamond surrounded by air?

(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 $30.0\text{\xc2\xb0}$ 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 parallel beam of light containing orange (610 nm) and violet (410 nm) wavelengths goes from fused quartz to water, striking the surface between them at a $60.0\text{\xc2\xb0}$ incident angle. What is the angle between the two colors in water?

A ray of 610-nm light goes from air into fused quartz at an incident angle of $55.0\text{\xc2\xb0}$. At what incident angle must 470 nm light enter flint glass to have the same angle of refraction?

A narrow beam of light containing red (660 nm) and blue (470 nm) wavelengths travels from air through a 1.00-cm-thick flat piece of crown glass and back to air again. The beam strikes at a $30.0\text{\xc2\xb0}$ incident angle. (a) At what angles do the two colors emerge? (b) By what distance are the red and blue separated when they emerge?

A narrow beam of white light enters a prism made of crown glass at a $45.0\text{\xc2\xb0}$ incident angle, as shown below. At what angles, ${\mathrm{\xce\xb8}}_{\text{R}}$ and ${\mathrm{\xce\xb8}}_{\text{V}}$, do the red (660 nm) and violet (410 nm) components of the light emerge from the prism?

## 1.7 Polarization

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

The angle between the axes of two polarizing filters is $45.0\text{\xc2\xb0}$. By how much does the second filter reduce the intensity of the light coming through the first?

Two polarizing sheets ${\text{P}}_{1}$ and ${\text{P}}_{2}$ are placed together with their transmission axes oriented at an angle $\mathrm{\xce\xb8}$ to each other. What is $\mathrm{\xce\xb8}$ when only $25\text{\%}$ of the maximum transmitted light intensity passes through them?

Suppose that in the preceding problem the light incident on ${\text{P}}_{1}$ is unpolarized. At the determined value of $\mathrm{\xce\xb8}$, what fraction of the incident light passes through the combination?

If you have completely polarized light of intensity $150\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$, what will its intensity be after passing through a polarizing filter with its axis at an $89.0\text{\xc2\xb0}$ angle to the lightâ€™s polarization direction?

What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity $1.00\phantom{\rule{0.2em}{0ex}}{\text{kW/m}}^{2}$ to reduce the intensity to $10.0\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$?

At the end of Example 1.7, it was stated that the intensity of polarized light is reduced to $90.0\text{\%}$ of its original value by passing through a polarizing filter with its axis at an angle of $18.4\text{\xc2\xb0}$ to the direction of polarization. Verify this statement.

Show that if you have three polarizing filters, with the second at an angle of $45.0\text{\xc2\xb0}$ to the first and the third at an angle of $90.0\text{\xc2\xb0}$ 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.)

Three polarizing sheets are placed together such that the transmission axis of the second sheet is oriented at $25.0\text{\xc2\xb0}$ to the axis of the first, whereas the transmission axis of the third sheet is oriented at $40.0\text{\xc2\xb0}$ (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?

In order to rotate the polarization axis of a beam of linearly polarized light by $90.0\text{\xc2\xb0}$, a student places sheets ${\text{P}}_{1}$ and ${\text{P}}_{2}$ with their transmission axes at $45.0\text{\xc2\xb0}$ and $90.0\text{\xc2\xb0}$, respectively, to the beamâ€™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 $30.0\text{\xc2\xb0}$ and $90.0\text{\xc2\xb0}$, respectively.

It is found that when light traveling in water falls on a plastic block, Brewsterâ€™s angle is $50.0\text{\xc2\xb0}$. What is the refractive index of the plastic?

What is Brewsterâ€™s angle for light traveling in water that is reflected from crown glass?

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