Objective Diameter of Telescope for Given Wavelength and Resolution

To find the value of the objective diameter $R$ of the telescope, we use Rayleigh's criterion for the angular resolution (limit of resolution) of a circular aperture.

The formula for the limit of angular resolution ($\theta$) of a telescope is given by: $\theta = \frac{1.22 \lambda}{R}$

where:

• $\theta$ is the angular resolution in radians,
• $\lambda$ is the wavelength of light,
• $R$ is the diameter of the objective lens.

From the problem statement, we are given:

• Wavelength of light, $\lambda = 500\text{ nm} = 500 \times 10^{-9}\text{ m} = 5 \times 10^{-7}\text{ m}$
• Angular resolution, $\theta = 5 \times 10^{-7}\text{ radian}$

Rearranging the formula to solve for the diameter $R$: $R = \frac{1.22 \lambda}{\theta}$

Substitute the given values into the equation: $R = \frac{1.22 \times \left(5 \times 10^{-7}\text{ m}\right)}{5 \times 10^{-7}\text{ rad}}$

$R = 1.22\text{ m}$

Converting the diameter $R$ from meters to centimeters: $R = 1.22 \times 100\text{ cm} = 122\text{ cm}$

Thus, the value of $R$ is $122\text{ cm}$, corresponding to Option B.