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Chapter 23: Problem 11

Sunlight strikes the surface of a lake. A diver sees the Sun at an angle of \(42.0^{\circ}\) with respect to the vertical. What angle do the Sun's rays in air make with the vertical?

Short Answer

Expert verified
Answer: The angle between the Sun's rays in air and the vertical is approximately 30.77°.

Step by step solution

01

Identify the given information

We are given the angle between the Sun's rays and the vertical in water, which is \(\theta_{2}=42.0^{\circ}\).
02

Find the index of refraction for air and water

The index of refraction for air (n1) is approximately 1.0003, and for water (n2) is approximately 1.333. These values are well-known and can be found in tables or literature.
03

Write down Snell's law of refraction

Snell's law states that: $$ n_1\sin(\theta_1)=n_2\sin(\theta_2) $$ Where \(n_1\) and \(n_2\) are the indices of refraction for air and water, respectively, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction in the media, respectively.
04

Solve for the angle of incidence, \(\theta_1\)

We are given \(n_1, n_2\), and \(\theta_2\), and we need to find \(\theta_1\). We can rearrange Snell's Law to solve for \(\theta_1\): $$ \theta_1 = \arcsin(\frac{n_2\sin(\theta_2)}{n_1}) $$ Now plug in the values we know: $$ \theta_1 = \arcsin(\frac{1.333\sin(42.0^{\circ})}{1.0003}) $$ Calculate the value of \(\theta_1\): $$ \theta_1 \approx 59.23^{\circ} $$
05

Calculate the angle between the Sun's rays in air and the vertical

We found the angle of incidence \(\theta_1\), which is the angle between the Sun's rays in air and the normal (perpendicular line) to the surface of the water. To find the angle between the Sun's rays and vertical, we need to calculate the complementary angle: $$ \text{Angle with the vertical} = 90^{\circ} - 59.23^{\circ} = 30.77^{\circ} $$ So, the angle between the Sun's rays in air and the vertical is approximately \(30.77^{\circ}\).

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Most popular questions from this chapter

In order to read his book, Stephen uses a pair of reading glasses. When he holds the book at a distance of \(25 \mathrm{cm}\) from his eyes, the glasses form an upright image a distance of \(52 \mathrm{cm}\) from his eyes. (a) Is this a converging or diverging lens? (b) What is the magnification of the lens? (c) What is the focal length of the lens?
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