Problem 5.4 — Prism Deviation at Critical Angle Incidence

Problem Statement

A ray hits a prism face at the critical angle $\theta_c$ for the glass–air interface. Prism angle $\Theta = 30°$, $n = 1.5$. Find the deviation of the emergent ray.

Given Information

• See problem statement for all given quantities.

Physical Concepts & Formulas

This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.

• See the step-by-step solution for the specific equations applied.
• All quantities are in SI units unless otherwise stated.

Step-by-Step Solution

Step 1 — Identify given quantities and set up the problem: $$\sin\theta_c = 1/n = 1/1.5 \implies \theta_c = 41.8°$$

Step 2 — Apply the relevant physical law or equation: Refraction at first face: $\sin r_1 = \sin\theta_c/n = 1/n^2 = 0.444 \implies r_1 = 26.4°$.

Step 3 — Solve algebraically for the unknown: At second face: $r_2 = \Theta – r_1 = 3.6°$, so $\sin\alpha_2 = n\sin r_2 = 0.094 \implies \alpha_2 = 5.4°$.

Step 4 — Substitute numerical values with units: $$\delta = (\theta_c – r_1)+(\alpha_2 – r_2) = 15.4°+1.8° \approx \boxed{17°}$$

Worked Calculation

$$\sin\theta_c = 1/n = 1/1.5 \implies \theta_c = 41.8°$$

$$\delta = (\theta_c – r_1)+(\alpha_2 – r_2) = 15.4°+1.8° \approx \boxed{17°}$$

$$\text{Numerical result} = \text{given expression substituted with values}$$

$$\sin\theta_c = 1/n = 1/1.5 \implies \theta_c = 41.8°$$

Refraction at first face: $\sin r_1 = \sin\theta_c/n = 1/n^2 = 0.444 \implies r_1 = 26.4°$.

At second face: $r_2 = \Theta – r_1 = 3.6°$, so $\sin\alpha_2 = n\sin r_2 = 0.094 \implies \alpha_2 = 5.4°$.

$$\delta = (\theta_c – r_1)+(\alpha_2 – r_2) = 15.4°+1.8° \approx \boxed{17°}$$

Answer

$$\delta = (\theta_c – r_1)+(\alpha_2 – r_2) = 15.4°+1.8° \approx \boxed{17°}$$

Physical Interpretation

The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.