Problem Statement
A single slit of width $b = 2.0$ mm is illuminated by a plane wave ($\lambda = 500$ nm) and the pattern is observed at $L = 3.0$ m. How many Fresnel half-period zones fit across the slit?
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: Width of one Fresnel zone (for plane wave, screen at $L$): $w = \sqrt{\lambda L}$.
Step 2 — Apply the relevant physical law or equation: $$w = \sqrt{500\times10^{-9}\times3.0} = \sqrt{1.5\times10^{-6}} = 1.22\times10^{-3}\text{ m} = 1.22\text{ mm}$$
$$N = \frac{b}{w} = \frac{2.0}{1.22} \approx \boxed{1.6\text{ zones (Fresnel regime)}}$$
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$w = \sqrt{500\times10^{-9}\times3.0} = \sqrt{1.5\times10^{-6}} = 1.22\times10^{-3}\text{ m} = 1.22\text{ mm}$$
$$N = \frac{b}{w} = \frac{2.0}{1.22} \approx \boxed{1.6\text{ zones (Fresnel regime)}}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
Width of one Fresnel zone (for plane wave, screen at $L$): $w = \sqrt{\lambda L}$.
$$w = \sqrt{500\times10^{-9}\times3.0} = \sqrt{1.5\times10^{-6}} = 1.22\times10^{-3}\text{ m} = 1.22\text{ mm}$$
$$N = \frac{b}{w} = \frac{2.0}{1.22} \approx \boxed{1.6\text{ zones (Fresnel regime)}}$$
Answer
$$N = \frac{b}{w} = \frac{2.0}{1.22} \approx \boxed{1.6\text{ zones (Fresnel regime)}}$$
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.
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