Problem Statement
A circular disk of diameter $2a = 1.0$ mm blocks the centre of a beam. The observation point is on the axis at distance $b = 1.0$ m. Wavelength $\lambda = 500$ nm. What is the intensity at the centre of the geometric shadow?
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: By Babinet’s principle, the diffraction pattern of the disk and the aperture of the same size are complementary: $U_{disk} + U_{aperture} = U_{free}$.
Step 2 — Apply the relevant physical law or equation: On the optical axis behind the disk, the amplitude equals that of the unobstructed wave (since the first Fresnel zone contributes amplitude $a_1$ and the disk blocks it, but by Babinet the remaining amplitude from all outer zones equals the free-field amplitude).
Step 3 — Solve algebraically for the unknown: $$\boxed{I_{axis} = I_0 \text{ (same as unobstructed beam — bright spot at centre of shadow)}}$$
Worked Calculation
$$\boxed{I_{axis} = I_0 \text{ (same as unobstructed beam — bright spot at centre of shadow)}}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\boxed{I_{axis} = I_0 \text{ (same as unobstructed beam — bright spot at centre of shadow)}}}$$
By Babinet’s principle, the diffraction pattern of the disk and the aperture of the same size are complementary: $U_{disk} + U_{aperture} = U_{free}$.
On the optical axis behind the disk, the amplitude equals that of the unobstructed wave (since the first Fresnel zone contributes amplitude $a_1$ and the disk blocks it, but by Babinet the remaining amplitude from all outer zones equals the free-field amplitude).
$$\boxed{I_{axis} = I_0 \text{ (same as unobstructed beam — bright spot at centre of shadow)}}$$
Answer
$$\boxed{I_{axis} = I_0 \text{ (same as unobstructed beam — bright spot at centre of shadow)}}$$
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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