Reflection, Refraction and Lenses: Question 4

Worked solution

Part (a): Calculating the refractive index

The refractive index of a material is defined using the angle of incidence in air, $i$, and the angle of refraction inside the material, $r$:

$n = \frac{\sin i}{\sin r}$

Substituting $i = 45°$ and $r = 30°$:

$n = \frac{\sin 45°}{\sin 30°} = \frac{0.7071}{0.5000}$

$n \approx 1.41$

Part (b): Diamond or glass imitation?

The calculated refractive index of the stone is about $1.41$.

Comparing this with the two reference values:

• Diamond: $n \approx 2.42$
• Glass: $n \approx 1.5$

The stone’s refractive index ($1.41$) is much closer to the value for glass than to the value for diamond. This suggests the stone is a glass imitation, not a genuine diamond.

Part (c): Calculating the critical angle

For light travelling from inside the stone out into air, the critical angle $c$ is related to the refractive index by:

$\sin c = \frac{1}{n}$

Using $n \approx 1.41$:

$\sin c = \frac{1}{1.41} \approx 0.7071$

$c = \sin^{-1}(0.7071)$

$c \approx 45°$

Part (d): Checking for total internal reflection at the next face

Total internal reflection occurs whenever the angle of incidence, measured inside the denser medium, is greater than the critical angle for that boundary.

At the second internal face, the angle of incidence is $50°$. Comparing this with the critical angle found in (c):

$50° > 45°$

Since the angle of incidence exceeds the critical angle, the ray does undergo total internal reflection at this face. It is reflected entirely back into the stone rather than refracting out.

Final answers

• (a) Refractive index $\approx \boxed{1.41}$
• (b) The stone is more likely a glass imitation, since $1.41$ is close to glass’s $1.5$ and far from diamond’s $2.42$
• (c) Critical angle $\approx \boxed{45°}$
• (d) Since $50° > 45°$, total internal reflection does occur at the second face