Question

For an aperture of $5 \times 10^{-3}\,\mathrm{m}$ and a monochromatic light of wavelength $\lambda$, the distance for which ray optics becomes a good approximation is $50\,\mathrm{m}$, then $\lambda =$

• A. $5000\,\mathring{A}$
• B. $6000\,\mathring{A}$
• C. $5400\,\mathring{A}$
• D. $6500\,\mathring{A}$

Hint:

Fresnel Distance ($Z_F$) defines the boundary between Ray Optics and Wave Optics. For distances $<Z_F$, diffraction effects are negligible (Ray Optics valid).

Answer 1

The Correct Option is A

Step 1: Concept (Fresnel Distance):
The distance up to which ray optics is a good approximation is called the Fresnel distance ($Z_F$). Formula: $Z_F = \frac{a^2}{\lambda}$ where $a$ is the aperture size and $\lambda$ is the wavelength.
Step 2: Calculation:
Given $Z_F = 50\,\mathrm{m}$, $a = 5 \times 10^{-3}\,\mathrm{m}$. \[ 50 = \frac{(5 \times 10^{-3})^2}{\lambda} \] \[ \lambda = \frac{25 \times 10^{-6}}{50} \] \[ \lambda = 0.5 \times 10^{-6}\,\mathrm{m} = 5 \times 10^{-7}\,\mathrm{m} \]
Step 3: Convert to Angstroms:
$1\,\mathring{A} = 10^{-10}\,\mathrm{m}$. \[ \lambda = 5000 \times 10^{-10}\,\mathrm{m} = 5000\,\mathring{A} \]