Suppose that you have a reflection diffraction grating with n = 120 lines per mi

Question

Suppose that you have a reflection diffraction grating with n = 120 lines per millimeter. Light from a sodium lamp passes through the grating and is diffracted onto a distant screen. Two visible lines in the sodium spectrum have wavelengths 498 nm and 569 nm. What is the angular separation Delta theta of the first maxima of these spectral lines generated by this diffraction grating? Express your answer in degrees to two significant figures. 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

let,

n=120 lines/mm

n=120000 lines/meter

and

slit width, d=1/n

wavelength,lambda1=498 nm

wavelength,lambda2=569 nm

a)

use,

d*sin(theta)=n*lambda

==>

if,

lambda1=498 nm

(1/120000)*sin(theta1)=1*498*10^-9

====> theta1=3.43 degrees

and

if,

lambda2=569 nm

(1/120000)*sin(theta2)=1*569*10^-9

====> theta2=3.92 degrees

now,

angular sepeartion, theta=3.92-3.43

theta=0.489 degrees

b)

order, m=2

L1=589 nm

L2=589.59 nm

Resolving power,

R=L1/(L2-L1)

and,

R=m*N

R=m*(d*n)

now,

L1/(L2-L1) = m*(d*n)

589/(589.59-589)=2*(d*120000)

===> d=4.16*10^-3 m

width of the grating, d=4.16 mm

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