91Ó°ÊÓ

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

$\mathrm{Δ}{\mathrm{θ}}_{hw}=\frac{\mathrm{λ}}{Nd\cos \mathrm{θ}}$ (half-width of the line )

Here, $\mathrm{λ}$is wavelength

d is ruling separation

N is number of rulings

Dispersion of a grating at an angle $\mathrm{θ}$is given by

$\frac{\mathrm{Δ}\mathrm{θ}}{\mathrm{Δ}\mathrm{λ}}=\frac{m}{d\cos \mathrm{θ}}$

Here, m is order,

d is grating space and

$\mathrm{Δ}\mathrm{λ}$is wavelength difference.

The path length difference is

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

Here $\mathrm{λ}$ is wavelength.

(a)

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

$\mathrm{Δ}{\mathrm{θ}}_{hw}=\frac{\mathrm{λ}}{Nd\cos \mathrm{θ}}$ (half-width of the line $\mathrm{θ}$)

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 $\mathrm{θ}$is given by

$\frac{\mathrm{Δ}\mathrm{θ}}{\mathrm{Δ}\mathrm{λ}}=\frac{m}{d\cos \mathrm{θ}}$

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

$d\sin \mathrm{θ}=m\mathrm{λ}$

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