Question

An optical engineer is calibrating a spectrometer that uses a diffraction grating to separate light in order of increasing wavelength

(λ_A,

λ_B,

and

λ_C).

He observes three distinct first-order spectral lines at the following respective angles

θ_m

(where m denotes order).

θ₁ = 12.8°, θ₁ = 13.9°, θ₁ = 15.3°

(a)

If the grating has 3,740 grooves per centimeter, what wavelength (in nm) describes each of these spectral lines?

at θ₁ = 12.8°

λ_A

= nmat θ₁ = 13.9°

λ_B

= nmat θ₁ = 15.3°

λ_C

= nm

(b)

At what angle (in degrees) would each of these lines be found in the second-order spectrum?

for λ_A

θ₂

= °for λ_B

θ₂

= °for λ_C

θ₂

= °

Answer 1

line spacing on grating, d = 1/3740 cm

= 10^-2/3740 m

= 2.67*10^-6 m

a) for theta1 = 12.8 degrees

use, d*sin(theta1) = 1*lamda

lamdaA = d*sin(theta1)

= 2.67*10^-6*sin(12.8)

= 5.91*10^-7 m

= 591 nm <<<<<<<---------------Answer

for theta1 = 13.9 degrees
lamdaB = d*sin(theta1)

= 2.67*10^-6*sin(13.9)

= 6.41*10^-7 m

= 641 nm <<<<<<<---------------Answer

for theta1 = 15.3 degrees
lamdaC = d*sin(theta1)

= 2.67*10^-6*sin(15.3)

= 7.04*10^-7 m

= 704 nm <<<<<<<---------------Answer

b)

for lamdaA = 591 nm

d*sin(theta2) = 2*lamda2

sin(theta2) = 2*lamda2/d

= 2*591*10^-9/(2.67*10^-6)

= 0.4427

theta2 = sin^-1(0.4427)

= 26.3 degrees <<<<<<<<<<-----------------Answer

for lamdaB = 641 nm

d*sin(theta2) = 2*lamda2

sin(theta2) = 2*lamda2/d

= 2*641*10^-9/(2.67*10^-6)

= 0.4801

theta2 = sin^-1(0.4801)

= 28.7 degrees <<<<<<<<<<-----------------Answer

for lamdaC = 704 nm

d*sin(theta2) = 2*lamda2

sin(theta2) = 2*lamda2/d

= 2*704*10^-9/(2.67*10^-6)

= 0.5273

theta2 = sin^-1(0.5273)

= 31.8 degrees <<<<<<<<<<-----------------Answer