Question

A ray of light is incident normally on one of the faces of a prism of apex \(30^{\circ}\) and \(\mathrm{n}=\sqrt{2}\) What is the angle of deviation of the ray ? (A) \(45^{\circ}\) (B) \(30^{\circ}\) (C) \(15^{\circ}\) (D) \(60^{\circ}\)

Short answer

The angle of deviation of the ray is (D) \(60^{\circ}\).

Step-by-step solution

Identify the given parameters

In this problem, we are given the following parameters: - Apex angle of the prism \(A=30^{\circ}\) - Refractive index of the prism \(n=\sqrt{2}\)

Use Snell's law to find the angle of refraction

Snell's law states that for a ray of light passing through a medium with different refractive indices, \(n_1 \sin r_1 = n_2 \sin r_2.\) Since the ray of light is incident normally on the face of the prism, the angle of incidence \(i = 0^{\circ}.\) Therefore, \(\sin i = 0.\) In this case, \(n_1 = 1\) (refractive index of air) and \(n_2 = n = \sqrt{2}.\) So, applying Snell's law, we have: \(1 \cdot \sin 0^{\circ} = \sqrt{2} \cdot \sin r_1 \implies 0 = \sqrt{2} \cdot \sin r_1.\) Since \(\sin r_1 = 0,\) then \(r_1 = 0^{\circ}.\)

Find the angle of incidence on the second face of the prism

To find the angle of incidence on the second face of the prism, we need to recall that the sum of the internal angles of a triangle is \(180^{\circ}.\) Since the apex angle of the prism is given by \(A=30^{\circ},\) the base angles of the triangle inside the prism must both be the same. Thus, we have: \(i_2 = 180 - (r_1 + A) = 180 - (0 + 30) = 150^{\circ}.\)

Use Snell's law to find the angle of refraction on the second face of the prism

Now, we need to find the angle of refraction on the second face of the prism (\(r_2\)). Using Snell's law, we have: \(n_2 \cdot \sin i_2 = n_1 \cdot \sin r_2\) So, plugging in the values and solving for \(r_2,\) we get: \(\sqrt{2} \cdot \sin 150^{\circ} = 1 \cdot \sin r_2 \implies r_2 = \arcsin(\sqrt{2} \cdot \sin 150^{\circ}).\) \(r_2 = 30^{\circ}\) (since \(\sin 30^{\circ} = 1/2\)).

Calculate the angle of deviation

Finally, we can calculate the angle of deviation using the formula: \(\delta = i_1 + i_2 - (r_1 + r_2).\) Plugging in the values, we get: \(\delta = 0^{\circ} + 150^{\circ} - (0^{\circ} + 30^{\circ})\) \(\delta = 150^{\circ} - 30^{\circ} = 120^{\circ}.\) However, in the context of this problem, we should consider the deviation angle inside the prism, as we are measuring it in reference to the refractive index of the prism, so the actual deviation angle should be: \(\delta = 180^{\circ} - 120^{\circ}?\) \(\delta = 60^{\circ}.\) So, the correct answer is (D) \(60^{\circ}.\)