69. A grating with lines uniformly spaced at 775 nm is illuminated by a sodium discharge lamp.

For what range of wavelength will there only be three principal lines generated?

Only two diffraction orders + 1 and - 1 are possible

Explanation:

The diffraction net is a system with many uniformly spaced lines, which is described by the expression

d sin θ = m λ m = 1, 2, 3, ...

Where d is the spread between the lines, λ the wavelength and m is an integer that corresponds to the order of diffraction, all wavelengths

For this case the spacing is

d = 775 nm

The sodium emission lines are 589.00 nm and 589.59 nm

With this data we can calculate how many diffraction orders are visible

θ = sin⁻¹ (m λ / d)

m = 1

λ1 = 589.00 nm

θ1 = sin⁻¹ (1 589/775)

θ1 = 49.46 °

λ2 = 589.59 nm

θ2 = sin⁻¹ ((1 589.59 / 775)

θ2 = 49.53 °

m = 2

λ1 = 589.00 nm

θ1 = sin⁻¹ (2 589/775)

θ1 = sin⁻¹ (1.5)

This order is not possible because the sin cannot be greater than 1

m = - 1

λ1 = 589 nm

θ1 = sin⁻¹ (-1 589/775)

θ1 = - 49.46

θ = sin⁻¹ (-1 589.59/775)

λ2 = 589.59 nm

θ2 = - 49.53 °

m = - 2

λ1 = 589 nm

θ1 = sin-1 (-2 589/775)

This order does not occur because the breast cannot be greater than 1

Only two diffraction orders + 1 and - 1 are possible