Problem 5.37 — Thin Film Interference (Reflected Light)

Problem Statement

A thin soap film ($n = 1.33$) is illuminated by white light at normal incidence. The film appears bright for $\lambda = 500$ nm in reflected light. Find the minimum thickness of the film.

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: For constructive interference in reflected light (with one phase reversal at top surface):

Step 2 — Apply the relevant physical law or equation: $$2nt = \left(m+\frac{1}{2}\right)\lambda, \quad m = 0, 1, 2, \ldots$$

Step 3 — Solve algebraically for the unknown: Minimum thickness ($m = 0$):

Step 4 — Substitute numerical values with units: $$t_{min} = \frac{\lambda}{4n} = \frac{500}{4\times1.33} = \frac{500}{5.32} \approx \boxed{94\text{ nm}}$$

Worked Calculation

$$2nt = \left(m+\frac{1}{2}\right)\lambda, \quad m = 0, 1, 2, \ldots$$

$$t_{min} = \frac{\lambda}{4n} = \frac{500}{4\times1.33} = \frac{500}{5.32} \approx \boxed{94\text{ nm}}$$

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

For constructive interference in reflected light (with one phase reversal at top surface):

$$2nt = \left(m+\frac{1}{2}\right)\lambda, \quad m = 0, 1, 2, \ldots$$

Minimum thickness ($m = 0$):

$$t_{min} = \frac{\lambda}{4n} = \frac{500}{4\times1.33} = \frac{500}{5.32} \approx \boxed{94\text{ nm}}$$

Answer

$$t_{min} = \frac{\lambda}{4n} = \frac{500}{4\times1.33} = \frac{500}{5.32} \approx \boxed{94\text{ nm}}$$

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.