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Chapter 23: Problem 10

Sunlight strikes the surface of a lake at an angle of incidence of \(30.0^{\circ} .\) At what angle with respect to the normal would a fish see the Sun?

Short Answer

Expert verified
Answer: The fish sees the Sun at an angle of approximately 22.1掳 with respect to the normal.

Step by step solution

01

Understand Snell's Law

Snell's Law states that the ratio of the sine of the angle of incidence (\(\theta_{1}\)) to the sine of the angle of refraction (\(\theta_{2}\)) is equal to the ratio of the refractive indices of the two materials: $$ \frac{\sin{\theta_{1}}}{\sin{\theta_{2}}} = \frac{n_{2}}{n_{1}} $$ In this case, \(n_{1}\) is the refractive index of air (approximated to 1.00), \(n_{2}\) is the refractive index of water (approximately 1.33), and \(\theta_{1}\) is the angle of incidence given in the problem.
02

Calculate the angle of refraction

We can rearrange Snell's Law to solve for the angle of refraction \(\theta_{2}\): $$ \theta_{2} = \arcsin \left( \frac{n_{1}\sin{\theta_{1}}}{n_{2}} \right) $$ Now, we can substitute the known values into this formula to calculate the angle of refraction: $$ \theta_{2} = \arcsin \left( \frac{1.00\sin{30.0^{\circ}}}{1.33} \right) $$
03

Evaluate the expression

Using a calculator, we find that: $$ \theta_{2} \approx \arcsin \left( \frac{0.5}{1.33} \right) \approx 22.1^{\circ} $$
04

Find the angle with respect to the normal

The angle of refraction we just calculated is the angle between the sunlight and the normal. Since the fish sees the Sun at this angle of refraction, the angle with respect to the normal is: $$ \theta_{\text{fish}} = \theta_{2} \approx 22.1^{\circ} $$ So, the fish sees the Sun at an angle of approximately \(22.1^{\circ}\) with respect to the normal.

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Most popular questions from this chapter

The focal length of a thin lens is \(-20.0 \mathrm{cm} .\) A screen is placed \(160 \mathrm{cm}\) from the lens. What is the \(y\) -coordinate of the point where the light ray shown hits the screen? The incident ray is parallel to the central axis and is \(1.0 \mathrm{cm}\) from that axis.
A scuba diver in a lake aims her underwater spotlight at the lake surface. (a) If the beam makes a \(75^{\circ}\) angle of incidence with respect to a normal to the water surface, is it reflected, transmitted, or both? Find the angles of the reflected and transmitted beams (if they exist). (b) Repeat for a \(25^{\circ}\) angle of incidence.
(a) Sunlight reflected from the still surface of a lake is totally polarized when the incident light is at what angle with respect to the horizontal? (b) In what direction is the reflected light polarized? (c) Is any light incident at this angle transmitted into the water? If so, at what angle below the horizontal does the transmitted light travel?
A dentist holds a small mirror \(1.9 \mathrm{cm}\) from a surface of a patient's tooth. The image formed is upright and 5.0 times as large as the object. (a) Is the image real or virtual? (b) What is the focal length of the mirror? Is it concave or convex? (c) If the mirror is moved closer to the tooth, does the image get larger or smaller? (d) For what range of object distances does the mirror produce an upright image?
A glass block \((n=1.7)\) is submerged in an unknown liquid. A ray of light inside the block undergoes total internal reflection. What can you conclude concerning the index of refraction of the liquid?
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