### Additional Problems

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.

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?

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 $45.0\text{\xb0}$, and you observe the angle of refraction to be $40.3\text{\xb0}$. What is the index of refraction of the substance and its likely identity?

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 $\left(\text{\Delta}x\right),$ given that the incident angle is $40.0\text{\xb0}$ and the glass is 1.00 cm thick.

Considering the previous problem, show that ${\theta}_{3}$ is the same as it would be if the second medium were not present.

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

Light reflected at $55.6\text{\xb0}$ from a window is completely polarized. What is the window’s index of refraction and the likely substance of which it is made?

(a) Light reflected at $62.5\text{\xb0}$ 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?

If ${\theta}_{\text{b}}$ is Brewster’s angle for light reflected from the top of an interface between two substances, and ${\theta}_{\text{b}}^{\text{\u2032}}$ is Brewster’s angle for light reflected from below, prove that ${\theta}_{\text{b}}+{\theta}_{\text{b}}^{\text{\u2032}}=90.0\text{\xb0}$.

**Unreasonable results** Suppose light travels from water to another substance, with an angle of incidence of $10.0\text{\xb0}$ and an angle of refraction of $14.9\text{\xb0}$. (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?

**Unreasonable results** Light traveling from water to a gemstone strikes the surface at an angle of $80.0\text{\xb0}$ and has an angle of refraction of $15.2\text{\xb0}$. (a) What is the speed of light in the gemstone? (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?

Suppose you put on two pairs of polarizing sunglasses with their axes at an angle of $15.0\text{\xb0}$. 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.

(a) On a day when the intensity of sunlight is $1.00\phantom{\rule{0.2em}{0ex}}{\text{kW/m}}^{2}$, 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 $20.0\text{\xb0}$. Assuming the sunlight is unpolarized and the polarizers are $100\text{\%}$ efficient, what is the initial rate of heating of the water in $\text{\xb0}\text{C/s}$, 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.