Question

An underwater diver sees the sun 59 degrees above horizontal.

How high is the sun above the horizon to a fisherman in a boat above the diver?

Express your answer using two significant figures.
φ= _____________degrees

Answer 1

Concepts and reason

The concept required to solve the given problem is Snell’s law.

Consider the law and substitute the values in the expression. Finally find the angle of sun that makes with the horizon observed by the fisherman in the boat.

Fundamentals

Snell’s law:

When the light travels from one medium to another, it gets deviated. This phenomenon is called refraction. It depends upon the angle as well as the media that are travelled by the light.

Snell’s law gives the relation between the angles of incidence, angle of refraction and the refractive indices of the media. This can be expressed as,

${n_1}\sin {\theta _1} = {n_2}\sin {\theta _2}$

Here, ${\theta _1}$ is the angle of incidence, ${\theta _2}$ is the angle of refraction, and ${n_1},{n_2}$ are the refractive indices of the media 1 and 2 respectively.

Angle of incidence:

The angle under water is,

$\begin{array}{c}\\{\theta _1} = 90^\circ - 59^\circ \\\\ = 31^\circ \\\end{array}$

This can be taken as the angle of incidence of the light.

Angle

Consider the expression for Snell’s law,

${n_1}\sin {\theta _1} = {n_2}\sin {\theta _2}$

Here, ${\theta _1}$ is the angle of incidence, ${\theta _2}$ is the angle of refraction, and ${n_1},{n_2}$ are the refractive indices of the media 1 and 2 respectively.

Here, the first medium is water and the second medium is air.

Substitute $31^\circ$ for angle of incidence, 1.33 for ${n_1},$ and 1 for ${n_2}$ .

The expression becomes,

$\begin{array}{c}\\1 \times \sin {\theta _2} = 1.33 \times \sin 31^\circ \\\\\sin {\theta _2} = 0.685\\\\{\theta _2} = {\sin ^{ - 1}}\left( {0.685} \right)\\\\ = 43.23^\circ \\\end{array}$

Angle made by the Sun with the horizon relative to the fisherman in the boat,

$\begin{array}{c}\\\theta = 90^\circ - 43.23^\circ \\\\ = 46.76^\circ \\\\ \approx 47^\circ \,\,\,\left( {{\rm{Two significant figures}}} \right)\\\end{array}$

Ans:

The angle between the Sun and the fisherman relative to the horizon is $47^\circ$ .