17. A plane transmission grating has 15000 lines per inch. Find (a) the resolving power ofthe grating, and (b) the smallest wavelength difference that can be resolved with alight of wavelength 6000 Å in the second order.
Question
- A plane transmission grating has 15000 lines per inch. Find (a) the resolving power ofthe grating, and (b) the smallest wavelength difference that can be resolved with alight of wavelength 6000 Å in the second order.
Solution
(a) The resolving power of a grating is given by the formula:
R = Nm
where N is the total number of lines on the grating and m is the order of the spectrum.
First, we need to find the total number of lines on the grating. We know that there are 15000 lines per inch. Let's convert this to lines per meter (since we usually work in SI units).
1 inch = 2.54 cm = 0.0254 m
So, N = 15000 lines/inch * 1/0.0254 m/inch = 590551.18 lines/m
For the first part of the question, we are looking for the resolving power in the second order (m=2).
So, R = Nm = 590551.18 * 2 = 1181102.36
(b) The smallest wavelength difference that can be resolved (Δλ) is given by the formula:
Δλ = λ/R
where λ is the wavelength of light.
We are given that λ = 6000 Å. We need to convert this to meters:
1 Å = 10^-10 m
So, λ = 6000 * 10^-10 m = 6 * 10^-7 m
Now we can find Δλ:
Δλ = λ/R = 6 * 10^-7 m / 1181102.36 = 5.08 * 10^-13 m
So, the smallest wavelength difference that can be resolved is 5.08 * 10^-13 m.
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