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Found in: Page 1108

### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# 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

1. Half-width decreases.
2. Separation will remain same.
3. The lines will remain in place.
See the step by step solution

## Step 1: The given data

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

## Step 2: Concept and Formula used

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

$\Delta {\theta }_{hw}=\frac{\lambda }{Nd\mathrm{cos}\theta }$ (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

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

Here, m is order,

d is grating space and

$\Delta \lambda$ is wavelength difference.

The path length difference is

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

Here $\lambda$ is wavelength.

## Step 3: Change in half-widths

(a)

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

$\Delta {\theta }_{hw}=\frac{\lambda }{Nd\mathrm{cos}\theta }$ (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.

## Step 4: Determine change in separation lines

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

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

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

So separation will remain same.

## Step 5: Determine shift in lines

(c)

The path length difference is

$d\mathrm{sin}\theta =m\lambda$

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.