Question

(a) Figure 36-34a shows the lines produced by diffraction gratingsA and B using light of the same wavelength; the lines are of the same order and appear at the same angles $\theta$. Which grating has the greater number of rulings? (b) Figure 36-34b shows lines of two orders produced by a single diffraction grating using light of two wavelengths, both in the red region of the spectrum. Which lines, the left pair or right pair, are in order with greater m? Is the center of the diffraction pattern located to the left or to the right in(c) Fig. 36-34a andd) Fig. 36-34b?

Short answer

1. Diffraction grating Ahas a larger number of rulings.
2. The left pair is in order with greater m.
3. The center of the diffraction pattern is located to the left.
4. The center of the diffraction pattern is on the right.

Step-by-step solution

The data given

Given figure 36-34a shows the lines produced by diffraction gratingsA and B using light of the same wavelength; the lines are of the same order and appear at the same angles$\theta .$

Figure 36-34b shows lines of two orders produced by a single diffraction grating using light of two wavelengths, both in the red region of the spectrum.

Concept and Formula used

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$\theta$ )

Here, $\lambda$is wavelength

d is ruling separation

N is number of rulings

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.

(a) Determine grating with greater number of rulings

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-width of the line $\theta$)

Here, half-width is inversely related to the number of rulings N.

Therefore, the smaller the half-widths$\Delta {\theta }_{hw}$,larger the number of rulings.

From the figure, it is clear that grate A has smaller half-widths.

Therefore, diffraction grating A has the larger number of rulings.

(b) Determine pair in order with greater m

The dispersion of a grating at an angle is given by

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

This implies larger orders produce lines with larger spacing.

Observe the spacing between each line. The left pair has more space between them.

So, the left pairis in order with greater m.

(c) Determine position of center of diffraction

It is known that the angle (as measured from the center) where the light appears is proportional to the wavelength.

Since red light has a longer wavelength than blue light, the red light must be at a greater angle from the center of the diffraction pattern.

Therefore, the center of the diffraction pattern is located to the left.

(d) Determine position of center of diffraction

In part (b) we found that the pair of lines on the left are in the order with the greater m

Pairs of lines in the order with the greaterm are farther from the center of the diffraction pattern.

Therefore, the center of the diffraction pattern is on the right.