Question

A horizontal ray is incident on one of the refracting faces of a prism of angle $8^\circ$. The ray strikes a vertical plane mirror after refraction as shown in figure. Find the angle (in degree) by which the mirror should be rotated in order to make the reflected ray antiparallel to incident ray.

(Take R.I. of material of prism = 5/4)

• A. The angle of deviation of the ray by the prism is $6^\circ$.
• B. The angle of deviation of the ray by the prism is $12^\circ$.
• C. The angle of deviation of the ray by the prism is $87^\circ$.
• D. The angle of deviation of the ray by the prism is $3^\circ$.

Answer 1

The angle of deviation of the ray by the prism is $6^\circ$.

Answer 2

The incident ray is horizontal, let's assume it's along the positive x-axis ($0^\circ$). The emergent ray from the prism is shown to be $6^\circ$ below the horizontal, so its angle is $-6^\circ$. The deviation caused by the prism is the difference between the emergent ray's direction and the incident ray's direction: $6^\circ - 0^\circ = 6^\circ$. This is the angle by which the ray is deviated downwards.

The ray incident on the mirror is at $-6^\circ$ with respect to the horizontal. We want the reflected ray to be antiparallel to the incident ray on the prism. Since the incident ray on the prism is at $0^\circ$, the reflected ray should be at $180^\circ$.

Let the angle of the normal to the mirror with the horizontal be $\alpha$. The mirror is initially vertical, meaning its normal is horizontal ($\alpha_{initial} = 0^\circ$). When a ray incident at angle $\theta_{inc}$ is reflected by a mirror with normal at angle $\alpha$, the reflected ray makes an angle $\theta_{ref} = 2\alpha - \theta_{inc}$ with the horizontal.

We have $\theta_{inc} = -6^\circ$ and we want $\theta_{ref} = 180^\circ$. Substituting these values: $180^\circ = 2\alpha_{final} - (-6^\circ)$ $180^\circ = 2\alpha_{final} + 6^\circ$ $2\alpha_{final} = 180^\circ - 6^\circ = 174^\circ$ $\alpha_{final} = 87^\circ$

The angle by which the mirror should be rotated is the change in the angle of its normal: Rotation angle = $\alpha_{final} - \alpha_{initial} = 87^\circ - 0^\circ = 87^\circ$.

The question asks for the angle by which the mirror should be rotated, which is $87^\circ$. However, the provided options seem to be related to the deviation by the prism. Based on the figure and the problem description, the emergent ray from the prism is at $-6^\circ$ relative to the horizontal incident ray. Therefore, the angle of deviation of the ray by the prism is $6^\circ$.