Question

The diameter of the objective of a telescope is 3.6 m. The limit of resolution of the telescope for a light of wavelength 540 nm is:

• A. \( 1.22 \times 10^{-7} \) rad
• B. \( 1.83 \times 10^{-7} \) rad
• C. \( 0.61 \times 10^{-7} \) rad
• D. \( 3.76 \times 10^{-7} \) rad

Hint:

To find the limit of resolution of an optical instrument, use the Rayleigh criterion: \[ \theta = \frac{1.22 \lambda}{D} \] where \( \lambda \) is the wavelength of light and \( D \) is the diameter of the aperture.

Answer 1

The Correct Option is B

Step 1: Understanding the Rayleigh Criterion. The limit of resolution (θ) of a telescope is given by Rayleigh's criterion: \[ \theta = \frac{1.22 \lambda}{D} \] where: - \( \lambda \) is the wavelength of light (540 nm = \( 540 \times 10^{-9} \) m), - \( D \) is the diameter of the telescope’s objective (3.6 m), - The factor \( 1.22 \) is derived from diffraction theory. 

Step 2: Substituting the values. \[ \theta = \frac{1.22 \times 540 \times 10^{-9}}{3.6} \] \[ \theta = \frac{658.8 \times 10^{-9}}{3.6} \] \[ \theta = 1.83 \times 10^{-7} \text{ rad} \] Final Answer: \[ \boxed{1.83 \times 10^{-7} \text{ rad}} \]