Problem 5.63 — Double Slit Diffraction Pattern

Problem Statement

In a double-slit experiment, slit width $b = 0.08$ mm, slit separation $d = 0.25$ mm, $\lambda = 600$ nm. How many interference fringes lie within 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: Central diffraction envelope: first minima at $\sin\theta = \lambda/b$. Number of interference maxima within: $m_{max} = d/b = 0.25/0.08 = 3.125$.

Step 2 — Apply the relevant physical law or equation: So $m = 0, \pm1, \pm2, \pm3$ lie inside, but $m = \pm3$ coincides with the diffraction minimum.

Step 3 — Solve algebraically for the unknown: $$\boxed{N = 5\text{ fringes (central + 2 on each side, excluding the missing } m=\pm3 \text{ orders)}}$$

Worked Calculation

$$\boxed{N = 5\text{ fringes (central + 2 on each side, excluding the missing } m=\pm3 \text{ orders)}}$$

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

$$\boxed{\boxed{N = 5\text{ fringes (central + 2 on each side, excluding the missing } m=\pm3 \text{ orders)}}}$$

Central diffraction envelope: first minima at $\sin\theta = \lambda/b$. Number of interference maxima within: $m_{max} = d/b = 0.25/0.08 = 3.125$.

So $m = 0, \pm1, \pm2, \pm3$ lie inside, but $m = \pm3$ coincides with the diffraction minimum.

$$\boxed{N = 5\text{ fringes (central + 2 on each side, excluding the missing } m=\pm3 \text{ orders)}}$$

Answer

$$\boxed{N = 5\text{ fringes (central + 2 on each side, excluding the missing } m=\pm3 \text{ orders)}}$$

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.