Problem 5.2 — Lateral Shift of Ray in a Glass Plate

Problem Statement

A ray of light falls on a glass plate of thickness $h = 1.5$ cm at angle of incidence $\theta = 60°$. Find the lateral shift of the ray after passing through the plate. Refractive index $n = 1.7$.

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: $$\sin r = \frac{\sin 60°}{1.7} = \frac{0.866}{1.7} = 0.510 \implies r = 30.7°$$

Step 2 — Apply the relevant physical law or equation: Path length inside the plate: $l = h/\cos r = 1.5/0.860 = 1.744$ cm.

Step 3 — Solve algebraically for the unknown: Lateral shift (perpendicular to incident ray):

Step 4 — Substitute numerical values with units: $$d = l\sin(\theta – r) = 1.744\,\sin(29.3°) = 1.744 \times 0.489 \approx \boxed{0.85\text{ cm}}$$

Worked Calculation

$$\sin r = \frac{\sin 60°}{1.7} = \frac{0.866}{1.7} = 0.510 \implies r = 30.7°$$

$$d = l\sin(\theta – r) = 1.744\,\sin(29.3°) = 1.744 \times 0.489 \approx \boxed{0.85\text{ cm}}$$

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

$$\sin r = \frac{\sin 60°}{1.7} = \frac{0.866}{1.7} = 0.510 \implies r = 30.7°$$

Path length inside the plate: $l = h/\cos r = 1.5/0.860 = 1.744$ cm.

Lateral shift (perpendicular to incident ray):

$$d = l\sin(\theta – r) = 1.744\,\sin(29.3°) = 1.744 \times 0.489 \approx \boxed{0.85\text{ cm}}$$

Answer

$$d = l\sin(\theta – r) = 1.744\,\sin(29.3°) = 1.744 \times 0.489 \approx \boxed{0.85\text{ cm}}$$

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.