What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete first-order spectrum for visible light

14,300 lines per cm

Explanation:

Given that

The wavelength of visible light ranges from 400 nm to 400nm

It is possible to find the maximum number of lines per cm. This maximum number of lines per cm is reciprocal of the least distance separating two adjacent slits, using the following equation.

mλ = dsin (θ), where

m = order of diffraction.

λ = wavelength of the incident light.

d = distance between the centers of the two slits.

θ = angle of diffraction of the mth order.

In order to find the least separation that allows the observation of one complete order of spectrum of the visible region, we make use the maximum or highest wavelength of the visible region that we are given, which is 700 nm.

d = mλ / sin (θ)

Again, we need the distance d to be the smallest, so sin (θ) must be the greatest, and for sin (θ) to be the greatest, it has to be equal to 1. Using the longest wavelength is the best idea because when the smallest wavelength is used the longest wavelength would not be diffracted.

d = mλ / sin (θ)

d = 1 * 700nm / 1

d = 700 nm

Next, we say that

n = 1/d

n = 1 / 700 x

n = 1, 430,000 lines per m, converting to cm, we have n = 14,300 lines per cm