Question

The spectrum of light from heated up hydrogen gas has only a few wavelengths present. These are known as spectral lines. It includes a red line at 656 nm and a blue-violet line at 434 nm. What are the angular separations between these two spectral lines for all visible orders obtained with a diffraction grating that has 4770 grooves/cm? (In this problem assume that the light is incident normally on the grating.)

I'm having trouble with this question as I'm certain that i can find the angle separation for the first order separation (6.2886 degrees), but the question is asking for all visible orders and i just want to make sure that i'm doing it right.

Answer 1

width of each slit, d = 1/(4770*100) m

= 2.096*10^-6 m

let n is the maximum visible order.

we know,

d*sin(theta) = n*lamda

when n = maximum, theta = 90 degrees

so, n = d/lamda

= 2.096*10^-6/(656*10^-9)

= 3.2

so, 3rd order is the maximum order we can see.

for red light,
d*sin(theta_r) = 3*lamda_r

sin(theta_r) = 3*lamda_r/d

= 3*656*10^-9/(2.096*10^-6)

= 0.9389

theta_r = sin^-1(0.9389)

= 69.9 degrees

for blue light,
d*sin(theta_b) = 3*lamda_b

sin(theta_b) = 3*lamda_b/d

= 3*434*10^-9/(2.096*10^-6)

= 0.6212

theta_b = sin^-1(0.9389)

= 38.4 degrees

so, angular separation, delta_theta = theta_r - theta_b

= 69.9 - 38.4

= 31.5 degrees <<<<<<<------------------Answer