A 500 line/mm diffraction grating is illuminated by light of wavelength 510 nm. How many diffraction orders are seen, and what is the angle of each?

n = 3 total orders seen

n = 1, Q_1 = 15 degrees

n = 2, Q_2 = 31 degrees

n = 3, Q_3 = 50 degrees

Explanation:

Given:

- Number of grating lines per mm N = 500 line/mm

- Wavelength of the light λ = 510 nm

Find:

How many diffraction orders are seen, and what is the angle of each?

Solution:

- We know that the maximum angle of diffraction Q_m of the furthest bright fringe < Q = 90 degrees.

- We need to compute the nth bright fringe for which is approximated to 90 degrees.

- The angle of nth bright fringe is given by:

sin (Q_m) = n*λ*N

- Approximating Q_m ≈ 90 degrees.

sin (90) = n*λ*N

n = sin (90) / λ*N

n = 1 / ((510*10^-6*) 500

n = 3.9 orders

- Since, we knew that Q_m < 90 degrees, we will choose n = 3 as the maximum number of orders. That means 3 fringes above and 3 fringes below the central order are observed.

- So for n = 1

sin (Q_n) = n*λ*N

sin (Q_1) = 1 * (510*10^-9) * (500,000)

Q_1 = sin^-1 (0.255)

Q_1 = 15 degrees

- So for n = 2

sin (Q_n) = n*λ*N

sin (Q_2) = 2 * (510*10^-9) * (500,000)

Q_2 = sin^-1 (0.51)

Q_2 = 31 degrees

- So for n = 3

sin (Q_n) = n*λ*N

sin (Q_3) = 3 * (510*10^-9) * (500,000)

Q_3 = sin^-1 (0.765)

Q_3 = 50 degrees