Question

Beyond what distance, the ray optics is sufficiently valid when the aperture is 6 mm wide and the wavelength is 6000 \AA?

• A. 50 m
• B. 60 m
• C. 40 m
• D. 10 m

Hint:

Always convert all given units to SI units (meters) before performing calculations. Remember that \( 1 \text{ \AA} = 10^{-10} \text{ m} \) and \( 1 \text{ mm} = 10^{-3} \text{ m} \).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the distance beyond which the effects of diffraction become significant and ray optics is no longer a good approximation, which corresponds to the Fresnel distance (\( Z_F \)).

Step 2: Key Formula or Approach:
The Fresnel distance \( Z_F \) is given by the formula:
\[ Z_F = \frac{d^2}{\lambda} \]
where:
- \( d \) is the width of the aperture.
- \( \lambda \) is the wavelength of light.

Step 3: Detailed Explanation:
Given values:
- Aperture width, \( d = 6 \text{ mm} = 6 \times 10^{-3} \text{ m} \)
- Wavelength, \( \lambda = 6000 \text{ \AA} = 6000 \times 10^{-10} \text{ m} = 6 \times 10^{-7} \text{ m} \)
Substitute these values into the formula:
\[ Z_F = \frac{(6 \times 10^{-3} \text{ m})^2}{6 \times 10^{-7} \text{ m}} \]
\[ Z_F = \frac{36 \times 10^{-6}}{6 \times 10^{-7}} \]
\[ Z_F = 6 \times 10^1 = 60 \text{ m} \]
Thus, for distances greater than 60 m, diffraction spread exceeds the size of the aperture, and wave optics must be used instead of ray optics.

Step 4: Final Answer:
The distance beyond which ray optics is no longer valid (or up to which it is valid) is 60 m.