Question

A telescope of objective lens diameter 2 m uses light of wavelength 5000 angstrom for viewing stars. The minimum angular separation between two stars whose image just resolved by the telescope is:
A. \[4\times {{10}^{-4}}\] rad
B. \[40.25\times {{10}^{-6}}\] rad
C. \[0.31\times {{10}^{-6}}\] rad
D. \[5\times {{10}^{-3}}\] rad

Answer 1

Hint: In this question, we are asked to calculate the minimum angular separation between two stars whose image has just been resolved. We know the formula to minimum angular separation for two stars. We will be using this formula to calculate the resolving power. Two images of objects are said to be just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other.

Formula Used:
 \[\Delta \theta =\dfrac{1.22\lambda }{d}\]
Where,
\[\lambda \] is the wavelength
d is the diameter
\[\Delta \theta \] is the minimum angular separation

Complete step by step answer:
It is given that diameter of objective lens is 2 m and wavelength is 5000 angstrom i.e. \[5000\times {{10}^{-10}}m\]
From the formula,
\[\Delta \theta =\dfrac{1.22\lambda }{d}\]
After substituting the given values in the above equation
We get,
\[\Delta \theta =\dfrac{1.22\times 5000\times {{10}^{-10}}}{2}\]
On solving
We get,
\[\Delta \theta =0.3\times {{10}^{-6}}\] rad
Therefore, the correct answer is option C.

Note: According to Rayleigh criterion, images of two different objects are said to be just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other. The resolving power of an optical instrument is its ability to separate the images of two very close objects. If the two central maxima of the image overlap, the images are said to be unresolved.
A telescope is an optical instrument with lenses or curved mirrors used to observe the distant objects such as a star in outer space.