A diffraction grating 2 cm wide is just able to resolve sodium D-lines (589 nm and589.6 nm) in second order. Find the number of rulings per mm.

To solve this problem, we need to use the formula for the minimum resolvable wavelength difference in a diffraction grating, which is given by:

Δλ = λ / (m*N)

where:

• Δλ is the minimum resolvable wavelength difference
• λ is the wavelength of light
• m is the order of diffraction
• N is the total number of slits in the diffraction grating

We are given that Δλ = 589.6 nm - 589 nm = 0.6 nm, λ = 589 nm, and m = 2. We are asked to find N, the total number of slits.

First, we need to convert the given wavelengths from nm to m for consistency in units. So, λ = 589 nm = 589 x 10^-9 m and Δλ = 0.6 nm = 0.6 x 10^-9 m.

Substituting these values into the formula, we get:

0.6 x 10^-9 m = 589 x 10^-9 m / (2*N)

Solving for N, we get:

N = 589 x 10^-9 m / (2 * 0.6 x 10^-9 m) N = 491.67

However, N must be an integer as it represents the total number of slits, so we round up to get N = 492.

This is the total number of slits in the 2 cm wide grating. To find the number of rulings per mm, we divide by the width of the grating in mm:

Number of rulings per mm = N / width = 492 / 20 mm = 24.6 rulings/mm

So, the diffraction grating has approximately 25 rulings per mm.