Solved Examples on Resolving Power
Example 1: A space telescope’s primary mirror has a diameter of 3.2 m. Determine the angle between two just-resolvable point light sources. Assume the average wavelength of light as 550 nm.
Solution:
Given,
Diameter (D) = 3.2 m
Wavelength of light (λ) =550 nm = 550 x 10-9 m
Using Rayleigh’s criterion,
θ = 1.22 (λ/D)
θ = 1.22 (550 x 10-9 / 3.2)
= 2.097 x 10-7 m
Example 2: Assume that light of wavelength 400nm is coming from a star. The limit of resolution of a telescope whose objective has a diameter of 2.6m is.
Solution:
Given,
Diameter of Objective Telescope (D) = 2.6 m = 100 x 2.6 cm = 260 cm
The wavelength of light (λ) = 400 nm = 4 x 10-5 cm
Limit of Resolution of telescope,
dθ = 1.22λ / D = 1.22 x 4 x 10-5/ 260
= 1.877 x 10-7 radian
Example 3: The diameter of the objective of a telescope is 1.8m. Its resolving limit for the light of wavelength 3462 A, will be
Solution:
Given,
Diameter of objective telescope (D) = 1.8 m
Wavelength of light (λ) = 3462 A = 3462 x 10-10 m
Resolving limit (dθ) = 1.22λ / D
= 1.22 x 3462 x 10-10 / 1.8
= 2.346 x 10-7 rad.
Example 4: The limit of resolution of the microscope, if the numerical aperture of the microscope is 0.26, and the wavelength of light used is 900 nm, is
Solution:
Given,
Numerical Aperture (A) = 0.26
Wavelength of light (λ) = 900 nm
For a microscope, the limit of resolution is given by,
X = λ/2A
= 900 / (2 x 0.26)
= 1.73 μm
Example 5: Calculate the resolving power of the microscope if its numerical aperture is 0.12 and wavelength of light is 5000 A.
Solution:
Given,
Numerical Aperture (A) = 0.12
Wavelength of light (λ) = 5000 A
Resolving Power of Microscope (RP) = 2μsinθ / λ
= 2A / λ …(Since, μsinθ = Numerical Aperture)
= 2 x 0.12 / 5000 x 10-10
= 4.8 x 105 m-1
Resolving Power of a Microscope and Telescope
Wave optics, also known as Physical optics, deals with the study of various phenomena such as diffraction, polarization, interference, resolution, and other occurrences. Wave optics is the segment of optics that focuses on the study and behavior of light and its wave characteristics. Wave optics particularly describes the connection between waves and rays of light. According to the wave theory of light, light is a form of energy, it travels through a medium in the form of transverse wave motion. The speed of light traveling through a medium depends upon the nature of the medium.
Here, the approximation is carried out by using ray optics for the estimation of the field on a surface. Integrating ray-estimated field over a mirror, lens, or aperture for the calculation of the scattered or transmitted field also gets involved later on. Wave optics stands as a witness to the scientific study of understanding the nature of light. One is the particle nature of light and the other is the wave nature of light.