Resolving Power of Plane Transmission Grating

Definition of Resolving Power:
Resolving power is defined as the ability of an optical instrument to distinguish between two closely spaced objects. In other words, it is the ability of an optical instrument to show two closely spaced spectral lines as separate.

Formula to Calculate Resolving Power:
Resolving power can be calculated using the formula:
R = λ/Δλ

Where R is the resolving power, λ is the wavelength of light used, and Δλ is the smallest difference in wavelength that the instrument can distinguish.

Resolving Power of Plane Transmission Grating:
A plane transmission grating is a type of diffraction grating that is used in optics for separating light into its different colors. It consists of a flat surface with many parallel grooves or lines, which act as a diffraction grating.

Given that the plane transmission grating has 40000 lines, we can use the formula to calculate the resolving power of the grating in second order.

First, we need to determine the distance between the adjacent lines of the grating. This can be done using the formula:
d = 1/N

Where d is the distance between the lines, and N is the number of lines per unit length. In this case, N = 40000 lines/meter, so we get:
d = 1/40000 = 0.000025 m = 25 μm

Now, we can calculate the angle of diffraction for the second order using the formula:
sin θ = mλ/d

Where θ is the angle of diffraction, m is the order of diffraction (in this case, m = 2), λ is the wavelength of light used, and d is the distance between the lines of the grating.

For a wavelength of 5000 Å (5000 x 10^-10 m), we get:
sin θ = 2 x 5000 x 10^-10 / 25 x 10^-6 = 0.0002
θ = sin^-1 (0.0002) = 0.0115 radians

The resolving power of the grating in second order can now be calculated using the formula:
R = λ/Δλ = λ/(2d sin θ)

For a wavelength of 5000 Å, we get:
R = 5000 x 10^-10 / (2 x 25 x 10^-6 x sin 0.0115) = 4346

Therefore, the resolving power of the plane transmission grating in second order for a wavelength of 5000 Å is 4346.