Question

A physics instructor wants to project a spectrum of visible-light colors from 400 nm to 700 nm as part of a classroom demonstration. She shines a beam of white light through a diffraction grating that has 600 lines per mm, projecting a pattern on a screen 2.9 m behind the grating.

Required:
How wide is the spectrum corresponding to m=1?

Answer 1

Dr = 263 10⁻⁶ m

Step-by-step explanation:

The diffraction pattern for constructive interference is described by

a sin θ = m λ

in this it indicates that the order of diffraction is m = 1

Let's use a direct proportion rule to find the separation of two slits. If there are 600 lines in 1 me, what is the distance between 2 slits

a = 2 lines 1/600

a = 2/600

a = 3.33 10⁻³ mm = 3.33 10⁻⁴ cm

let's use trigonometry

tan θ = y / L

as the measured angles are small

tan θ = sin θ / cos θ sin θ

sin θ = y / L

we substitute

a y/L = λ

y = λ L / a

for λ = 400 10-9 m

I = 400 10⁻⁹ 2.9 / 3.33 10⁻³

i = 346.89 10⁻⁶ m

f

or λ = 700 nm

y_f = 700 10⁻⁻⁹ 2.9 / 3.33 10⁻³

y_f = 609.609 10⁻⁶ m

the separation of this spectrum

Δr = v_f - i

Dr = (609.609 - 346) 10 ⁻⁶

Dr = 263 10⁻⁶ m