Question

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.

• A. 61
• B. 122
• C. 244
• D. 305

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The resolving power of a telescope describes its ability to distinguish between two closely spaced distant objects. The angular resolution (limit of resolution) is determined by the Rayleigh criterion.
Step 2: Key Formula or Approach:
The angular resolution \(\Delta \theta\) of a telescope is given by the formula:
\[ \Delta \theta = \frac{1.22 \lambda}{D} \] where \(\lambda\) is the wavelength of the light and \(D\) is the diameter of the objective lens (denoted as \(R\) in this problem).
Step 3: Detailed Explanation:
We are given:
Wavelength, \(\lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m}\).
Angular resolution, \(\Delta \theta = 5 \times 10^{-7} \text{ radians}\).
Substitute the values into the formula to solve for the diameter \(R\):
\[ 5 \times 10^{-7} = \frac{1.22 \times 500 \times 10^{-9}}{R} \] Rearranging for \(R\):
\[ R = \frac{1.22 \times 500 \times 10^{-9}}{5 \times 10^{-7}} \] \[ R = \frac{610 \times 10^{-9}}{5 \times 10^{-7}} \] \[ R = \frac{6.1 \times 10^{-7}}{5 \times 10^{-7}} \] \[ R = 1.22 \text{ m} \] Convert the diameter into centimeters as requested by the question:
\[ R = 1.22 \times 100 \text{ cm} = 122 \text{ cm} \] Step 4: Final Answer:
The value of \(R\) is 122.