Suppose that you have a reflection diffraction grating with n = 140 lines per mi

Question

Suppose that you have a reflection diffraction grating with n= 140 lines per millimeter. Light from a sodium lamp passes through the grating and is diffracted onto a distant screen.

Part A

Two visible lines in the sodium spectrum have wavelengths 498 nm and 569 nm. What is the angular separation of the first maxima of these spectral lines generated by this diffraction grating?

Part B

How wide does this grating need to be to allow you to resolve the two lines 589.00 and 589.59 nanometers, which are a well known pair of lines for sodium, in the second order (m=2)?

Explanation / Answer

given data

n=140 lines/mm

wave lengths = 498 nm and 569 nm

PART A)

What is the angular separation

(0.001/140)*(cos )*d = (569-498)*(10)---- (1) and
(0.001/140)*(sin ) = 569*(10) ------------ (2)
Dividing (1) by (2), we get
(cot )*d = 71/569 = 0.125or
d = 0.125*tan -------------------- (3)
(2) gives = arc sin[0.000569*140] 4.569 degree or
tan = 0.0799 ------------------ (5), substituting in (3) we get
required d = 0.00997 or 0.01 degree

PART B)

Resolvance R = / ~= 1000.
Also R = mN, where m is the order and N is the number of slits.
Thus mN = 1000, and since m = 2, N = 500.
So width = 500/140 mm = 3571 m.

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