Dispersive Power of Grating

Dispersive Power of Grating:

The angular spacing between any two spectral lines is nothing but dispersion. Using plane diffraction grating, dispersion of spectral lines can be achieved.

The dispersive power of grating is defined as the ratio of the difference in the angle of diffraction of any two neighbouring spectral lines to the difference in the wavelengths between these spectral lines.

Dispersive power is also defined as the difference in the angle of diffraction per unit change in wavelength.

If θ1 and θ2 are angles of diffraction in a particular order (say ‘n’) for wavelengths λ1 and λ2 respectively.

Then, dθ/dλ = (θ1 – θ2)/(λ1 – λ2) is called dispersive power in that order.

We know that the nth order principal maximum for a wavelength ‘λ’ is-

(a + b) sin θ = nλ ……….(i)
where (a + b) is grating element.

Differentiating equation (i) and keeping (a + b) and n constants, we get-

(a + b) cos θ dθ = n dλ
⇒ dθ/dλ = n/[(a + b) cos θ] ……….(ii)

Here 1/(a + b) stands for the number of lines per unit length. Thus it is clear that dispersion is more for grating having a large number of lines on it and it is minimum for θ = 0.

Maximum Number of Order of Diffraction available with Grating:

We know that condition of principal maxima is-

(a + b) sin θ = nλ
∴ n = (a + b) sin θ/λ ……….(iii)
Thus for n to be nmax, sin θ = 1,
∴ nmax = (a + b)/λ ……….(iv)

∴ If (a + b) < λ, nmax < 1 ⇒ nmax = 0;
(a + b) < 2λ, nmax < 2 ⇒ nmax = 1 (i.e. n = 0 and n = 1)

Example- If (a + b) = 1.69 x 10-4 cm and λ = 5890 Å = 5890 x 10-8 cm
then, nmax = 2.78 < 3 ⇒ nmax = 2 (i.e. n = 0, 1 and 2)

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