Question

A Fraunhofer diffraction pattern is produced by a circular aperture of radius $0.05 \, \text{cm}$ at the focal plane of a convex lens of focal length $20 \, \text{cm}$. If the wavelength $\lambda = 5 \times 10^{-5} \, \text{cm}$, the radius of the first dark ring is:

• A. $1.22 \times 10^{-3} \, \text{cm}$
• B. $12.20 \times 10^{-3} \, \text{cm}$
• C. $12.20 \times 10^{-2} \, \text{cm}$
• D. $12.20 \, \text{cm}$

Hint:

When calculating the radius of the first dark ring for a circular aperture in diffraction phenomena, ensure to double the radius to find the diameter if only the radius is given. Also, remember that this calculation assumes the ideal theoretical conditions where the light source and observing screen are effectively at infinite distances, achieved in practice by the lens focusing.

Answer 1

The Correct Option is B

For a circular aperture, the radius of the first dark ring in the Fraunhofer diffraction pattern, often referred to as the Airy disk radius \( r_1 \), is given by: \[ r_1 = 1.22 \frac{\lambda f}{D}, \] where \( D \) is the diameter of the aperture. Given that the radius \( a = 0.05 \, \text{cm} \), the diameter \( D = 2a = 0.1 \, \text{cm} \). Substituting the given values: \[ r_1 = 1.22 \times \frac{5 \times 10^{-5} \, \text{cm} \times 20 \, \text{cm}}{0.1 \, \text{cm}} = 1.22 \times 10^{-2} \, \text{cm} = 12.20 \times 10^{-3} \, \text{cm}. \] This calculation confirms that the radius of the first dark ring matches option (b).