Question

You have 2 diffraction gratings, one with 2000 lines/cm, the other with 5000 lines/cm. You are also given a light source that emits 450 nm blue light and 530 nm green light at the same time. A. What is the line spacing for each grating? B. What is the difference in the angle of the 2nd order maximum for each wavelength if you use the 5000 line/cm grating?? C. Which color is closer to the center of the diffraction pattern? D. Would the 2000 lines/cm grating result in a larger, a smaller, or the same separation?

Answer 1

Given that

Two diffraction gratings, one with 2000lines/cm and the other with 5000lines/cm

A)

The line spacing between the each grating is given by

e+d =1/2000 =5*10^-4cm-1

e+d =1/5000=2*10^-4cm-1

B)

We know that in the plane transmission grating (e+d)sintheta =nlamda

Here e+d =e+d =1/5000=2*10^-4cm-1, n=2 , lamda1 =450nm =450*10^-9m

For the blue light sint (theta1) = n*lamda/(e+d) =2*450*10^-9m/2*10^-6m-1 =450*10^-3

theta1 =sin-1(0.45) =26.74degrees

Here e+d =e+d =1/5000=2*10^-4cm-1, n=2 , lamda2 =530nm =530*10^-9m

For the green light sint (theta2) = n*lamda/(e+d) =2*530*10^-9m/2*10^-6m-1 =530*10^-3

theta1 =sin-1(0.53) =32degrees

The angular separation is given by theta2-theta1 =32-26.74 =5.265degrees

C)

We know that color closer to the center of diffraction pattern will be blue light

D)

We know that in the plane transmission grating (e+d)sintheta =nlamda

Here e+d =e+d =1/2000=5*10^-4cm-1, n=2 , lamda1 =450nm =450*10^-9m

For the blue light sint (theta1) = n*lamda/(e+d) =2*450*10^-9m/5*10^-6m-1 =180*10^-3

theta1 =sin-1(0.18) =10.36degrees

Here e+d =e+d =1/2000=5*10^-4cm-1, n=2 , lamda2 =530nm =530*10^-9m

For the green light sint (theta2) = n*lamda/(e+d) =2*530*10^-9m/5*10^-6m-1 =212*10^-3

theta1 =sin-1(0.212) =12.23degrees

The angular separation is given by theta2-theta1 =12.23 -10.36 =1.87degrees

Therefore the angular separation will be larger for 5000lines/cm and smaller for 2000lies/cm