Question

Diameter of the objective of a telescope is 200 cm. What is the resolving power of the telescope? Take wavelength of light $ = 5000\mathop {\text{A}}\limits^0 $.
$
  {\text{A}}{\text{. 6}}{\text{.56}} \times {\text{1}}{{\text{0}}^6} \\
  {\text{B}}{\text{. 3}}{\text{.28}} \times {\text{1}}{{\text{0}}^5} \\
  {\text{C}}{\text{. 1}} \times {\text{1}}{{\text{0}}^6} \\
  {\text{D}}{\text{. 3}}{\text{.28}} \times {\text{1}}{{\text{0}}^6} \\
$

Answer 1

Hint: The resolving power of the telescope is given as the ratio of the diameter of the objective lens of the telescope to 1.22 times the wavelength of the light coming from the object and reaching the telescope.

Formula used:
The resolving power of a telescope is given as:
$R = \dfrac{D}{{1.22\lambda }}{\text{ }}...{\text{(i)}}$

where R signifies the resolving power of the telescope, D is the diameter of the objective lens used in the telescope while $\lambda $ signifies the wavelength of the light coming from the object and reaching the telescope.

Complete step-by-step answer:
Telescope is a device used to view objects which are at a very large distance from the observer. It has two lenses: the objective lens and the eyepiece.

The resolving power of an optical instrument can be defined as the ability of that instrument to produce just separate images of two objects that are viewed as very close to each other. For a telescope, the resolving power depends on the diameter of the objective and wavelength of the incident light.

We are given a telescope whose objective lens has following value of the diameter

$D = 200cm = 2m$

The wavelength of the light coming from the object and reaching the telescope is given as
$\lambda = 5000\mathop {\text{A}}\limits^0 = 5 \times {10^{ - 7}}m$

Now using the formula for resolving power of telescope given in equation (i) and substituting all the known values, we get

$
  R = \dfrac{D}{{1.22\lambda }} \\
   = \dfrac{2}{{1.22 \times 5 \times {{10}^{ - 7}}}} = 3.28 \times {10^6} \\
$

This is the required answer so option D is correct.

Note:
1. The formula for resolving power of the telescope is derived using the concept of diffraction. It describes how much the diffraction patterns of the two closely placed objects overlap each other in the lens of the microscope.
2. The fundamental difference between a microscope and a telescope is that in a telescope the light is coming from the object that is being viewed at a distance while in case of a microscope we shine light on an object kept very close to the lens of the microscope.