A telescope with objective diameter $$R$$ is used to observe a distant star emitting light of wavelength 500 nm, at a resolution of $$5 \times 10^{-7}$$ radian. The value of $$R$$ is _____ cm.

The minimum angular resolution of an optical telescope is governed by the Rayleigh criterion:

$$\theta_{\text{min}} = 1.22 \frac{\lambda}{D}$$

where
  • $$\theta_{\text{min}}$$ is the smallest resolvable angle (in radians),
  • $$\lambda$$ is the wavelength of light used, and
  • $$D$$ is the diameter of the objective aperture.

Given data:
$$\theta_{\text{min}} = 5 \times 10^{-7}\ {\text{rad}}$$
$$\lambda = 500\ \text{nm} = 500 \times 10^{-9}\ \text{m} = 5 \times 10^{-7}\ \text{m}$$

Rearrange the Rayleigh formula to find $$D$$:

$$D = 1.22 \frac{\lambda}{\theta_{\text{min}}}$$

Substitute the numerical values:

$$D = 1.22 \times \frac{5 \times 10^{-7}\ \text{m}}{5 \times 10^{-7}\ \text{rad}}$$

The factor $$5 \times 10^{-7}$$ cancels out in numerator and denominator, giving

$$D = 1.22\ \text{m}$$

Convert metres to centimetres:

$$1.22\ \text{m} = 1.22 \times 100\ \text{cm} = 122\ \text{cm}$$

Thus the required diameter of the telescope objective is $$R = 122\ \text{cm}$$.

Option B which is: 122