Question

A beam of light strikes a sheet of glass at an angle of 57.0deg. with the normal in air. You observe that red light makes an angle of 38.1 degrees with the normal in the glass, while violet light makes a 36.7 degree angle.

1.What are the indexes of refraction of this glass for these colors of light?
2.What are the speeds of red and violet light in the glass?

Answer 1

Concepts and reason

The concepts required to solve the given question is Snell’s law and the sped in the different media.

Initially, calculate the refractive index for the red light and the violet light by using the Snell’s law equation. Later, calculate the speed of the red light. Finally, calculate the speed of the violet light.

Fundamentals

The expression for Snell’s law is as follows:

$\frac{{{n_{\rm{r}}}}}{{{n_2}}} = \frac{{\sin {\theta _{\rm{i}}}}}{{\sin {\theta _{\rm{r}}}}}$

Here, ${n_{\rm{r}}}$ is the refractive index of red light, ${n_2}$ is the refractive index of air, ${\theta _{\rm{i}}}$ is the angle of incidence, and ${\theta _{\rm{r}}}$ is the angle of refraction.

The expression for Snell’s law is as follows:

$\frac{{{n_{\rm{v}}}}}{{{n_2}}} = \frac{{\sin {\theta _{\rm{i}}}}}{{\sin {\theta _{\rm{r}}}}}$

Here, ${n_{\rm{v}}}$ is the refractive index of red light, ${n_2}$ is the refractive index of air, ${\theta _{\rm{i}}}$ is the angle of incidence, and ${\theta _{\rm{r}}}$ is the angle of refraction.

The expression for the velocity of the red light in as follows:

${v_{\rm{r}}} = \frac{c}{{{n_{\rm{r}}}}}$

Here, c is the speed of light.

The expression for the velocity of the red light in as follows:

${v_{\rm{v}}} = \frac{c}{{{n_{\rm{v}}}}}$

Here, c is the speed of light.

(1)

Substitute $57^\circ$ for ${\theta _{\rm{i}}}$ , $38.1^\circ$ for ${\theta _{\rm{r}}}$ , and 1.00 ${n_2}$ in the equation $\frac{{{n_{\rm{r}}}}}{{{n_2}}} = \frac{{\sin {\theta _{\rm{i}}}}}{{\sin {\theta _{\rm{r}}}}}$ .

$\begin{array}{c}\\\frac{{{n_{\rm{r}}}}}{{1.00}} = \frac{{\sin 57^\circ }}{{\sin 38.1^\circ }}\\\\{n_{\rm{r}}} = \frac{{0.838}}{{0.617}}\\\\ = 1.36\\\end{array}$

Substitute $57^\circ$ for ${\theta _{\rm{i}}}$ , $36.7^\circ$ for ${\theta _{\rm{r}}}$ , and 1.00 ${n_2}$ in the equation $\frac{{{n_{\rm{v}}}}}{{{n_2}}} = \frac{{\sin {\theta _{\rm{i}}}}}{{\sin {\theta _{\rm{r}}}}}$ .

$\begin{array}{c}\\\frac{{{n_{\rm{v}}}}}{{1.00}} = \frac{{\sin 57^\circ }}{{\sin 36.7^\circ }}\\\\{n_{\rm{v}}} = \frac{{0.838}}{{0.597}}\\\\ = 1.40\\\end{array}$

(2)

Substitute $3 \times {10^8}{\rm{ m/s}}$ for c and 1.36 for ${n_{\rm{r}}}$ in the equation ${v_{\rm{r}}} = \frac{c}{{{n_{\rm{r}}}}}$ .

$\begin{array}{c}\\{v_{\rm{r}}} = \frac{{3 \times {{10}^8}{\rm{ m/s}}}}{{1.36}}\\\\ = 2.21 \times {10^8}{\rm{ m/s}}\\\end{array}$

Substitute $3 \times {10^8}{\rm{ m/s}}$ for c and 1.40 for ${n_{\rm{v}}}$ in the equation ${v_{\rm{v}}} = \frac{c}{{{n_{\rm{v}}}}}$ .

$\begin{array}{c}\\{v_{\rm{v}}} = \frac{{3 \times {{10}^8}{\rm{ m/s}}}}{{1.40}}\\\\ = 2.14 \times {10^8}{\rm{ m/s}}\\\end{array}$

Ans: Part 1

The refractive index of red light is 1.36 and the refractive index of the violet light is 1.40.

Part 2

The speed of the red light is equal to $2.21 \times {10^8}{\rm{ m/s}}$ and the speed of the violet light is equal to $2.14 \times {10^8}{\rm{ m/s}}$ .