Resolving Power of Grating

Resolving Power Definition:

The separation between two objects which are very close to each other is called resolution and the ability of an optical instrument to produce their separate patterns is known as resolving power.

According to Rayleigh’s criterion of the resolution, the two spectral lines (or two-point sources) are said to be just resolved when the principal maximum in the diffraction pattern of one falls over the 1st minimum in the diffraction pattern of the other and vice versa.

Let λ1 and λ2 be the wavelengths of two spectral lines resolved by the grating. Then if their wavelength difference is large, their principal maxima are separately visible and lines are well resolved (Fig. a).

Now when (λ2 – λ1) is small in such a way that the central maximum of λ, coincides with 1st minimum of λ2 and vice-versa, then the lines are said to be just resolved (Fig. b).

Resolving Power Diagram

Resolving Power of Grating:

Resolving Power of Grating is its ability to form two separate maxima of two wavelengths which are very close to each other. It is defined as the ratio of the wavelength of one of the spectral line to the difference in the wavelengths between them such that the two lines appear to be just resolved.

Thus, R.P. = λ/dλ
Resolving Power of Grating Diagram

(1) Consider incident light is having two spectral lines of wavelengths ‘λ’ and ‘λ + dλ’.

(2) Let ‘θ’ and ‘θ + dθ’ be their angles of diffraction respectively for the nth order as shown in the figure.

(3) Condition for nth principal maximum for λ is-

(a + b) sin θ = nλ ……….(i)

(4) Similarly nth principal maximum of (λ + dλ) is expressed as-

(a + b) sin (θ + dθ) = n (λ + dλ) ……….(ii)

(5) Now according to Rayleigh’s criterion, if nth principal maximum due to (λ + dλ) coincides on 1st minimum after nth principal maximum due to λ, then only two objects appear as separate.

(6) We know that condition for minima is-

(a + b) sin θ = mλ/N ……….(iii)
But m = 1, as it is 1st minimum.

(7) Thus at an angle ‘θ + dθ’ there is 1st minimum (λ/N) after nth principal maximum due to λ (i.e. nλ).

Therefore, we can write-

(a + b) sin (θ + dθ) = nλ + λ/N ……….(iv)
⇒ n (λ + dλ) = nλ + λ/N [using (ii) and (iv)]
⇒ n dλ = λ/N
⇒ λ/dλ = Resolving Power = nN ……….(v) which is required equation.

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