For the situation in Questions 9 and Fig. 1, if instead we increased the grating spacing, would (a) the half-widths 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. Gating spacing $d$ is increased.

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

The dispersion of a grating at an angle 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

localid="1663139636669" $d\sin \theta =m\lambda$, for $m=0, 1, 2, ...$(maxima lines)

Here $\lambda$is wavelength.

The 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 the width of the line )

Here, half-width is inversely related to grating spacing d.

So if the grating space increases, half the width decreases.

The dispersion of a grating at an angle is given by

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

Separation of lines is inversely related to grating space, so

So if the grating space increases, the separation of the lines decreases.

The path length difference is

$d\sin \theta =m\lambda$

Since the grating spacing d increases, the path length difference also increases, so lines will shift to the right.