Question

Find the value of the angle of emergence from the prism. Refractive index of the glass is $\sqrt{3}$.

• A. $60^{\circ}$
• B. $30^{\circ}$
• C. $45^{\circ}$
• D. $90^{\circ}$

Answer 1

The Correct Option is A

To solve this problem, we need to determine the angle of emergence for a light ray passing through a prism given that the refractive index of the glass is \(\sqrt{3}\).

We will use the concept of refraction through a prism and apply Snell's Law. Let's consider the following:

• A: Angle of the prism
• \(n\): Refractive index of the glass = \(\sqrt{3}\)
• \(\theta_i\): Angle of incidence
• \(\theta_r\): Angle of refraction inside the prism
• \(\theta_e\): Angle of emergence

We know from Snell's Law at the first interface:

\(n_1 \sin(\theta_i) = n \sin(\theta_r)\) (Equation 1)

Since the prism is symmetric and the light emerges symmetrically, for minimum deviation \(D\), the angle of incidence and angle of emergence are equal (\(\theta_i = \theta_e\)). The prism formula relates these angles and the refractive index as follows:

\(n = \frac{\sin((A+D)/2)}{\sin(A/2)}\)

Given that the angle of minimum deviation \(D\) for the prism with refractive index \(\sqrt{3}\) results in:

\(\sqrt{3} = \frac{\sin((A+60^\circ)/2)}{\sin(A/2)}\)

By substituting basic geometry and trigonometry, for a common equilateral prism where:

\(A = 60^\circ\), it gives \(\theta_r = 30^\circ\).

Hence, the angle of emergence will be equal to the angle of incidence:

\(\theta_{e} = 60^\circ\)

Thus, the correct option is:

\(60^{\circ}\)