Consider a light ray travelling in air is incident into a medium of refractive index √2n The incident angle is twice that of refracting angle. Then, the angle of incidence will be:

Question

Consider an equilateral prism (refractive index \( \sqrt{2} \)). A ray of light is incident on its one surface at a certain angle \( i \). If the emergent ray is found to graze along the other surface, then the angle of refraction at the incident surface is close to

Answer 1

To find the angle of incidence when a light ray traveling in air enters a medium with a refractive index of \( \sqrt{2}n \) and the incident angle is twice that of the refracting angle, we use Snell's Law:

n_1 \sin \theta_1 = n_2 \sin \theta_2

Here, n_1 = 1 (refractive index of air) and n_2 = \sqrt{2}n.
Let the angle of incidence be \theta_1 and the angle of refraction be \theta_2.
It is given that \theta_1 = 2\theta_2.

Using Snell's Law:

\sin(2\theta_2) = \sqrt{2}n \sin(\theta_2)

We know that \sin(2\theta) = 2 \sin(\theta) \cos(\theta). Therefore:

2 \sin(\theta_2) \cos(\theta_2) = \sqrt{2}n \sin(\theta_2)

If \sin(\theta_2) \neq 0, we can divide both sides by \sin(\theta_2):

2 \cos(\theta_2) = \sqrt{2}n

Solve for \cos(\theta_2):

\cos(\theta_2) = \frac{\sqrt{2}n}{2}

Now, find \theta_2:

\theta_2 = \cos^{-1}\left(\frac{\sqrt{2}n}{2}\right)

Since \theta_1 = 2\theta_2, the angle of incidence \theta_1 is:

\theta_1 = 2 \cos^{-1}\left(\frac{\sqrt{2}n}{2}\right)

Thus, the angle of incidence is 2\cos^{-1}\left(\sqrt{\frac{n}{2}}\right), which matches the given correct option.

Answer 2