Problem 5.61 — Single Slit: Width of Central Maximum

Problem Statement

A slit of width $b = 0.20$ mm is illuminated by light $\lambda = 550$ nm. A lens of focal length $f = 50$ cm focuses the pattern on a screen. Find the width of the central diffraction maximum.

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: The first minima occur at $\sin\theta_1 = \lambda/b$. The half-width of the central maximum on the screen is:

Step 2 — Apply the relevant physical law or equation: $$y_1 = f\tan\theta_1 \approx f\frac{\lambda}{b} = 0.50\times\frac{550\times10^{-9}}{0.20\times10^{-3}} = 0.50\times2.75\times10^{-3} = 1.375\text{ mm}$$

Step 3 — Solve algebraically for the unknown: Full width of central maximum:

Step 4 — Substitute numerical values with units: $$w = 2y_1 \approx \boxed{2.75\text{ mm}}$$

Worked Calculation

$$y_1 = f\tan\theta_1 \approx f\frac{\lambda}{b} = 0.50\times\frac{550\times10^{-9}}{0.20\times10^{-3}} = 0.50\times2.75\times10^{-3} = 1.375\text{ mm}$$

$$w = 2y_1 \approx \boxed{2.75\text{ mm}}$$

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

The first minima occur at $\sin\theta_1 = \lambda/b$. The half-width of the central maximum on the screen is:

$$y_1 = f\tan\theta_1 \approx f\frac{\lambda}{b} = 0.50\times\frac{550\times10^{-9}}{0.20\times10^{-3}} = 0.50\times2.75\times10^{-3} = 1.375\text{ mm}$$

Full width of central maximum:

$$w = 2y_1 \approx \boxed{2.75\text{ mm}}$$

Answer

$$w = 2y_1 \approx \boxed{2.75\text{ mm}}$$

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.