Question

Some distant star is to be observed by some telescope of diameter of objective lens \(a\), at an angular resolution of \(3.0 \times 10^{-7}\) radian. If the wavelength of light from the star reaching the telescope is 500 nm, the minimum diameter of the objective lens of the telescope is ________ cm.

Answer 1

Correct Answer: 203

Step 1: Understanding the Concept:
The angular resolution (or limit of resolution) of a telescope is the minimum angle between two distant objects such that they can be seen as distinct. This is limited by the diffraction of light through the circular aperture of the objective lens.
Step 2: Key Formula or Approach:
According to the Rayleigh criterion for a circular aperture: \[ \Delta \theta = \frac{1.22 \lambda}{a} \] where \(\Delta \theta\) is the angular resolution, \(\lambda\) is the wavelength, and \(a\) is the diameter of the objective.
Step 3: Detailed Explanation:
Given: \(\Delta \theta = 3.0 \times 10^{-7}\) rad, \(\lambda = 500 \text{ nm} = 5 \times 10^{-7}\) m. Rearranging for \(a\): \[ a = \frac{1.22 \lambda}{\Delta \theta} \] \[ a = \frac{1.22 \times 5 \times 10^{-7}}{3.0 \times 10^{-7}} \] \[ a = \frac{6.10}{3} = 2.0333... \text{ m} \] To convert meters to centimeters: \[ a = 2.0333 \times 100 = 203.33 \text{ cm} \] The nearest integer is 203.
Step 4: Final Answer:
The minimum diameter is 203 cm.