Suppose that you have a reflection diffraction grating with n = 105 lines per mi

Question

Suppose that you have a reflection diffraction grating with n= 105 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?

Express your answer in degrees to two significant figures.

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)?

Express your answer in millimeters to two significant figures.

Explanation / Answer

here,
w1 = 498 * 10^-9 m
w2 = 569 * 10^-9 m
slit seperation distance,d = 1/0.105

[part a] :

from sigle slit diffraction experiment we ahve :
SinA*d = m*wavelength

SinA = m*wavelength/d

case a:
SinA1 = (1*498 * 10^-9)/(1/0.105)
SinA1 = 5.23*10^-8 m
SinA1 = 0.523 nm
A1 = 31.53 degrees

Case B:
SinA2 = (1*569 * 10^-9)/(1/0.105)
SinA2 = 5.97*10^-8 m = 0.597 nm
A2 = arcSin(0.597)
A2 = 36.6

Del(A) = 36.6 - 31.53
Del(A) = 5.07 degrees

Part B:
Resolvance, R = /

R = 589 / (589.59-589)
R = 998.31

since
R = m*N
where,
m is the order
N is the number of slits.

Therefore
2*N = 998.31
N = 500

Therefore Width will be :
D = 500/105 mm
D = 4.76 mm
or
D = 476 m

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