Problem 5.107 — Thin Film: Transmitted vs Reflected

Problem Statement

A soap film ($n = 1.33$, $t = 380$ nm) is illuminated at normal incidence with white light. Which wavelength in the range 400–700 nm is most strongly transmitted?

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: Maximum transmission = minimum reflection. Destructive reflection at $2nt = m\lambda$ (this is where there would be constructive transmission).

Step 2 — Apply the relevant physical law or equation: $$\lambda = \frac{2nt}{m} = \frac{2\times1.33\times380}{m} = \frac{1011}{m}\text{ nm}$$

Step 3 — Solve algebraically for the unknown: $m=1$: $\lambda = 1011$ nm (IR). $m=2$: $\lambda = 505.5$ nm (green, visible).

Step 4 — Substitute numerical values with units: $$\boxed{\lambda \approx 506\text{ nm (green)} \text{ is most strongly transmitted}}$$

Worked Calculation

$$\lambda = \frac{2nt}{m} = \frac{2\times1.33\times380}{m} = \frac{1011}{m}\text{ nm}$$

$$\boxed{\lambda \approx 506\text{ nm (green)} \text{ is most strongly transmitted}}$$

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

Maximum transmission = minimum reflection. Destructive reflection at $2nt = m\lambda$ (this is where there would be constructive transmission).

$$\lambda = \frac{2nt}{m} = \frac{2\times1.33\times380}{m} = \frac{1011}{m}\text{ nm}$$

$m=1$: $\lambda = 1011$ nm (IR). $m=2$: $\lambda = 505.5$ nm (green, visible).

$$\boxed{\lambda \approx 506\text{ nm (green)} \text{ is most strongly transmitted}}$$

Answer

$$\boxed{\lambda \approx 506\text{ nm (green)} \text{ is most strongly transmitted}}$$

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.