Figure shows a red line and a green line of the same order in the pattern produced by a diffraction grating. If we increased the number of rulings in the grating – say, by removing tape that had covered the outer half of the rulings – would (a) the half-widhts of the lines and (b) the separation of the lines increase, decrease, or remain the same? (c) Would the lines shift to the right, shift to the left, or remain in place

Given a red line and a green line of the same order in the pattern produced by a diffraction grating.

Numbers of rulings are increased in the grating – say, by removing tape that had covered the outer half of the rulings

Half-width of any other line depends on its location relative to the central axis and is

$\Delta {\theta }_{hw}=\frac{\lambda }{Nd\cos \theta }$ (half-width of the line )

Here, $\lambda$is wavelength

d is ruling separation

N is number of rulings

Dispersion of a grating at an angle $\theta$is given by

$\frac{\Delta \theta }{\Delta \lambda }=\frac{m}{d\cos \theta }$

Here, m is order,

d is grating space and

$\Delta \lambda$is wavelength difference.

The path length difference is

$d\sin \theta =m\lambda$, for $m=0, 1, 2, ...$(maxima lines)

Here $\lambda$ is wavelength.

(a)

Half-width of any other line depends on its location relative to the central axis and is

$\Delta {\theta }_{hw}=\frac{\lambda }{Nd\cos \theta }$ (half-width of the line $\theta$)

Here, half-width are inversely related to number of sits.

So if the number of slits increases, half width decreases.

Dispersion of a grating at an angle $\theta$is given by

$\frac{\Delta \theta }{\Delta \lambda }=\frac{m}{d\cos \theta }$

It can be seen that amount of slits, N, is independent of separation of the lines.

So separation will remain same.

(c)

The path length difference is

$d\sin \theta =m\lambda$

Since the distance between slits, d, and the order, m, and the wavelength, $\lambda$, all the factors will remain the same for each light, the position of lines will also remain the same.