A diffraction grating, which has 4000 lines/cm is used at normal incidence. Calculatethe dispersive power of the grating in the third order spectrum in the wavelengthregion 5000Å.
Question
A diffraction grating, which has 4000 lines/cm is used at normal incidence. Calculatethe dispersive power of the grating in the third order spectrum in the wavelengthregion 5000Å.
Solution
The dispersive power (P) of a diffraction grating is given by the formula:
P = dλ/dθ
where dλ is the change in wavelength and dθ is the change in the angle of diffraction. However, this formula is not easy to use directly. Instead, we can use the formula for the angular dispersion of a grating, which is given by:
dθ/dλ = m/Nd
where m is the order of the spectrum, N is the number of lines per unit length on the grating, and d is the spacing between the lines on the grating.
First, we need to find the value of d. Since there are 4000 lines per cm on the grating, we have:
N = 4000 lines/cm = 4000 * 10^2 lines/m = 4 * 10^5 lines/m
Therefore, the spacing between the lines is:
d = 1/N = 1/(4 * 10^5 m) = 2.5 * 10^-6 m
Now we can calculate the angular dispersion:
dθ/dλ = m/Nd = 3/(4 * 10^5 * 2.5 * 10^-6 m) = 3/10^6 rad/Å
Therefore, the dispersive power of the grating in the third order spectrum in the wavelength region 5000Å is 3/10^6 rad/Å.
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