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 resolving power of a telescope determines its ability to distinguish between two closely spaced distant objects. The minimum angular resolution is limited by diffraction at the circular aperture of the objective lens, dictated by the Rayleigh criterion.
Step 2: Key Formula or Approach:
The formula for the minimum angular resolution $\Delta \theta$ is:
$\Delta \theta = \frac{1.22 \lambda}{a}$
where $\lambda$ is the wavelength of light and $a$ is the diameter of the objective lens.
Step 3: Detailed Explanation:
Given values:
Angular resolution $\Delta \theta = 3.0 \times 10^{-7}\text{ rad}$.
Wavelength $\lambda = 500\text{ nm} = 500 \times 10^{-9}\text{ m} = 5 \times 10^{-7}\text{ m}$.
Substitute these into the Rayleigh criterion formula to solve for $a$:
$3.0 \times 10^{-7} = \frac{1.22 \times (5 \times 10^{-7})}{a}$.
Rearrange to isolate $a$:
$a = \frac{1.22 \times 5 \times 10^{-7}}{3.0 \times 10^{-7}}$.
The $10^{-7}$ terms cancel out:
$a = \frac{1.22 \times 5}{3.0} = \frac{6.10}{3.0}\text{ m}$.
$a \approx 2.0333\text{ m}$.
Convert the diameter into centimeters:
$a \text{ (in cm)} = 2.0333 \times 100\text{ cm} = 203.33\text{ cm}$.
Rounding to the nearest integer gives 203.
Step 4: Final Answer:
The minimum diameter of the objective lens is 203 cm.