Question

The index of refraction for diamond is 2.42. For a diamond in the air (index of refraction$ = 1.00$), what is the smallest angle that a light ray inside the diamond can make with a normal and completely reflect back inside the diamond (the critical angle)?
A) ${90^ \circ }$
B) ${45^ \circ }$
C) ${68^ \circ }$
D) ${66^ \circ }$
E) ${24^ \circ }$

Answer 1

Hint:The process in which the rays of light travel from a more optically denser medium to a less optically denser medium is known as total internal reflection of light. In total internal reflection, the angle of reflection is greater than the critical angle.

Complete step by step solution:
Step I:
In total internal reflection when the incident ray travels through an optically denser medium (diamond) gets reflected back at the boundary between air and diamond, instead of getting passed, the ray of light gets refracted.
The formula to find the angle, when the refractive index of one medium is less than the refractive index of another medium can be written as
${\theta _c} = {\sin ^{ - 1}}(\dfrac{{{n_2}}}{{{n_1}}})$
Where ${\theta _c}$is the critical angle
${n_2}$is the refractive index of air
and ${n_1}$is the refractive index of diamond.

Step II:
Here ${n_2} = {n_{air}} = 1.00$
And ${n_1} = {n_{diamond}} = 2.42$
Therefore, substituting all the given values in the formula, the critical angle will be
${\theta _c} = {\sin ^{ - 1}}(\dfrac{{1.00}}{{2.42}})$
${\theta _c} = {\sin ^{ - 1}}(0.413)$
${\theta _c} = {24^ \circ }$
Step III:
Therefore, it is clear that the smallest angle that a light ray inside the diamond can make with a normal and completely reflect back inside the diamond will be ${24^ \circ }$.

$ \Rightarrow $Option E is the right answer.

Note:It is to be noted that in total internal reflection, the critical angle is the angle of incidence beyond which when the rays of light pass from an optically denser medium to less denser medium will not undergo refraction. But they will be reflected totally from the surface. In other words, critical angle is the smallest angle of incidence due to which total internal reflection occurs.