Answer:
y = 6 10⁻² m
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
