**Answer:**

N = 221.4 lines / mm

**Explanation:**

**Given:**

- The wavelength of the source λ = 670 nm

- Distance of the grating from screen B = 40.0 cm

- The distance of first bright fringe from central order P = 6.0 cm

**Find:**

How many lines per millimeter does this grating have?

Solution:

- The derived results from Young's experiment that relates the order of bright fringes about the central order is given by:

sin (Q) = n*λ*N

Where,

n is the order number 0, 1 , 2, 3 , ....

λ is the wavelength of the light source

Q is the angle of sweep respective fringe from central order

N is the number of lines/mm the grating has

- We will first compute the length along which the light travels for the first bright fringe:

L^2 = P^2 + B^2

L^2 = 40^2 + 6^2

L^2 = 1636

L = 40.45 cm

- Now calculate the sin(Q) that the fringe makes with the central order:

sin (Q) = P / L

sin (Q) = 6 / 40.45

- Now we will use the derived results:

N = sin(Q) / n*λ

Where, n = 1 - First order

Plug values in N = (6 / 40.45) / (670 *10^-6)

N = 221.4 lines / mm