An unpolarised light is incident at an interface of two dielectric media having refractive indices of $2$ (incident medium) and $2\sqrt{3}$ (refracted medium) respectively. To satisfy the condition that reflected and refracted rays are perpendicular to each other, the angle of incidence is ___.

Question

The exit surface of a prism with refractive index $n$ is coated with a material having refractive index $\dfrac{n}{2}$. When this prism is set for minimum angle of deviation, it exactly meets the condition of critical angle. The prism angle is ___.

Answer 1

To determine the angle of incidence such that the reflected and refracted rays are perpendicular, we can utilize the concept of Brewster's angle. When the reflected and refracted rays are perpendicular, the angle of incidence is known as Brewster's angle.

Brewster's angle, denoted as $ \theta_B $, can be calculated using the formula:

\[\tan \theta_B = \frac{n_2}{n_1}\]

Where:

$ n_1 $ is the refractive index of the first medium (incident medium).
$ n_2 $ is the refractive index of the second medium (refracted medium).

In this problem:

$ n_1 = 2 $
$ n_2 = 2\sqrt{3} $

Substituting the given values into the Brewster's angle formula:

\[\tan \theta_B = \frac{2\sqrt{3}}{2} = \sqrt{3}\]

The angle whose tangent is $ \sqrt{3} $ is $ 60^\circ $, as:

\[\tan 60^\circ = \sqrt{3}\]

Therefore, the angle of incidence $ \theta_B $ is $ 60^\circ $.

Thus, the correct answer is: $ 60^\circ $

Answer 2