Question

Light from two lasers is incident on an opaque barrier with a single slit of width 4.0 x 10-5 m. One laser emits light of wavelength 4.0 x 10-7 m and the other is 6.0 x 10-7 m. A screen to view the light intensity pattern is 2.0 m behind the barrier. What is the distance from the center of the pattern to the nearest completely dark spot (dark for both colors)?

Answer 1

y = 6 10⁻² m

Step-by-step explanation:

This is a diffraction exercise which is described by the expression

a sin θ = m λ

we can use trigonometry to find the distance from the inside of the screen to the dark point (y)

tan θ = y / L

angles are very small in diffraction experiments, so we can approximate

tan θ = sin θ /cos θ = sin θ

sin θ = y / L

substituting

a (y / L) = m λ

Let's find the points where the intensity becomes zero

y = m L λ / a

λ₁ = 4.0 10⁻⁷ m

m = 1

y = 1 2.0 4.0 10⁻⁷/ 4.0 10⁻⁵

y = 2 10⁻² m

m = 2

y = 4 10⁻² m

λ₂ = 6.0 10⁻⁷ m

m = 1

y = 1 2.0 6.0 10⁻⁷ / 4.0 10⁻⁵

y = 3 10⁻² m

m = 2

y = 6 10⁻² m

if we want a point where the two colors are dark, we set the two expressions equal

y₁ = y₂

m₁ L λ₁ / a = m₂ L λ₂ / a

m₁/m₂ = λ₂/λ₁

m₁ / m₂ = 6 10⁻⁷ / 4 10⁻⁷

m₁ / m₂ = 1.5

since the quantities m must be integers, the smallest relation that the relation fulfills is

m₁ = 3

m₂ = 2

the distance for this destructive interference is

y = 3 2 4.0 10⁻⁷ / 4.0 10⁻⁵

y = 6 10⁻² m

this is the first point where the minimum of the two wavelengths coincide