Light traveling from water to a gemstone strikes the surface at an angle of 80.0º and has an angle of refraction of 15.2º.

(a) What is the speed of light in the gemstone?

(b) What is unreasonable about this result?

(c) Which assumptions are unreasonable or inconsistent?

At the point of refraction, the angle formed by a refracted beam and a line drawn normal to the interface between two media.

a)

Let us solve the given problem.

• The angle of incidence, ${\theta }_1={80}^{°}$
• The angle refraction, ${\theta }_2=15.2^{°}$
• Formula used:

From Snell's law of refraction,

$n_1\sin {\theta }_1=n_2\sin {\theta }_2..............\left(1\right)$

Here,

• n₁Is the refractive index of the medium 1.
• ${\theta }_1$Is the angle of incidence.
• $n_2$Is the refractive index of the medium 2.
• ${\theta }_2$Is the angle of refraction.

The relation between the refractive index and the speed is,

$n=\frac{c}{v}..............\left(2\right)$

Here,

nis the refractive index of the medium.

cis the speed of light.

vis the speed of the light in that medium.

Rearrange the equation (1) for the refractive index of the second medium.

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

Substitute 1.33 for$n_1,80.0^{°}$for ${\theta }_1$and $15.2^{°}$for ${\theta }_2$in the above equation.

$$\begin{gathered} n_2=\frac{1.33\sin \left(80.0^{°}\right)}{\sin \left(15.2^{°}\right)} \\ =5.00 \end{gathered}$$

Rearrange equation (2) for speed of light in the medium.

$v=\frac{c}{n}$

Substitute 5.00 for n and$3\times {10}^{8}m/s$ for c in equation (2).

$$\begin{gathered} v=\frac{3\times {10}^{8}m/s}{5.00} \\ =0.6\times {10}^{8}m/s \end{gathered}$$

Thus, the speed of the light in the gemstone is $0.6\times {10}^{8}m/s$.

b)

Consider theindex of refraction of the second substance in part a).

Therefore, the low magnitude of the speed of light is unreasonable.

The calculated light speed in the gemstone is$0.6\times {10}^{8}m/s$.When compared to the speed of light, this is very little difference. The predicted light speed in gemstones is 19% a fraction of the speed of light.

Therefore, the gemstone's light speed is unreasonably slow.

c)

We are stating the assumptions which are unreasonable.

When compared to the angle of refraction, the angle of incidence is extremely big. As a result, the angle of incidence and angle of refraction is quite different. As a result, the assumption about the angle difference is unreliable.

Therefore, the assumption about the angle difference is incorrect.