Find Minimum Diameter of Objective Lens of Telescope

To find the minimum diameter of the objective lens of the telescope, we use Rayleigh's criterion for the limit of angular resolution of a circular aperture:

$\theta = \frac{1.22 \lambda}{a}$

where:

• $\theta$ is the angular resolution in radians,
• $\lambda$ is the wavelength of light reaching the telescope,
• $a$ is the minimum diameter of the objective lens.

Given Data:

• Angular resolution, $\theta = 3.0 \times 10^{-7} \text{ rad}$
• Wavelength of light, $\lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m} = 5 \times 10^{-7} \text{ m}$

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Calculation:

Rearranging Rayleigh's formula to solve for $a$:

$a = \frac{1.22 \lambda}{\theta}$

Substitute the given values into the equation:

$a = \frac{1.22 \times 5 \times 10^{-7} \text{ m}}{3.0 \times 10^{-7}}$

$a = \frac{6.1 \times 10^{-7}}{3.0 \times 10^{-7}} \text{ m}$

$a = \frac{6.1}{3.0} \text{ m} \approx 2.0333 \text{ m}$

Converting the diameter into centimeters ($\text{cm}$):

$a = 2.0333 \times 100 \text{ cm} \approx 203.33 \text{ cm}$

Rounding to the nearest integer gives:

$a \approx 203 \text{ cm}$