Dispersive Power of Grating

Dispersive power of a 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 wavelength between the two spectral lines. It can also be defined as the diffraction in the angle of diffraction per unit change in wavelength. The diffraction of the n th order principal maximum for a wavelength λ is given by the equation,

(a + b) sin θ = nλ                      (i)

Differentiating this equation with respect to θ and λ (a + b is constant and n is constant in a given order)

(a + b) cos θ dθ = n dλ



In equation (ii) dθ/dλ is the dispersive power, n is the order of the spectrum, N’ is the number of lines per cm of the grating surface and θ is the angle of diffraction for the n th order principal maximum of wavelength λ.

From equation (ii), it is clear, that the dispersive power of the grating is (1) directly proportional to the number of lines per cm and (3) inversely proportional to cos θ. Thus, the angular spacing of any two spectral lines is double in the second order spectrum in comparison to the first order. Secondly, the angular dispersion of the lines is more with a grating having larger number of lines per cm. thirdly, the angular dispersion is minimum when θ = 0. If the value of θ is not large the value of cos can be taken as unity approximately and the influence of the factor cos θ in the equation (ii) can be neglected.

Neglecting the influence of cos θ, it is clear that the angular dispersion of any two spectral lines (in a particular order) is directly proportional to the difference in wavelength between the two spectral lines. A spectrum of this type is called a normal spectrum.

If the linear spacing of two spectral lines of wavelength λ and λ + dλ is dx in the focal plane of the telescope objective or the photographic plate, then,

dx = ƒ dθ

where ƒ is the focal length of the objective. The linear dispersion



The linear dispersion is useful in studying the photographs of a spectrum.

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