Problem Statement
A circular aperture of diameter $D = 1.0$ mm is illuminated by a plane wave ($\lambda = 550$ nm). A lens of focal length $f = 100$ cm focuses the light. Find the radius of the Airy disk (first dark ring).
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Circular motion requires a centripetal force directed toward the centre, providing the centripetal acceleration $a_c = v^2/r = \omega^2 r$. This force is not a new type of force — it is always the resultant of real forces (tension, normal force, friction, gravity) directed inward. At the minimum speed for maintaining contact, the normal force drops to zero.
- $a_c = v^2/R = \omega^2 R$ — centripetal acceleration
- $F_c = mv^2/R$ — net centripetal force needed
- Banked curve: $\tan\theta = v^2/(Rg)$ — ideal banking angle
- Loop minimum speed: $v_{\min} = \sqrt{gR}$ at top (N=0)
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: $$r_{Airy} = 1.22\frac{\lambda f}{D} = 1.22\times\frac{550\times10^{-9}\times1.0}{1.0\times10^{-3}} = 1.22\times5.5\times10^{-4}$$
$$= 6.71\times10^{-4}\text{ m} \approx \boxed{0.67\text{ mm}}$$
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$r_{Airy} = 1.22\frac{\lambda f}{D} = 1.22\times\frac{550\times10^{-9}\times1.0}{1.0\times10^{-3}} = 1.22\times5.5\times10^{-4}$$
$$= 6.71\times10^{-4}\text{ m} \approx \boxed{0.67\text{ mm}}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$r_{Airy} = 1.22\frac{\lambda f}{D} = 1.22\times\frac{550\times10^{-9}\times1.0}{1.0\times10^{-3}} = 1.22\times5.5\times10^{-4}$$
$$= 6.71\times10^{-4}\text{ m} \approx \boxed{0.67\text{ mm}}$$
Answer
$$= 6.71\times10^{-4}\text{ m} \approx \boxed{0.67\text{ mm}}$$
Physical Interpretation
The centripetal force is not a ‘new’ force but the net inward resultant of real forces. If that resultant falls below $mv^2/r$, the object cannot maintain circular motion and will fly outward — this is the critical condition for minimum speed problems.
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