Coherent light with wavelength 610 nm passes through two very narrow slits, and the interference pattern is observed on a screen a distance of 3.00 m from the slits. The first-order bright fringe is a distance of 4.84 mm from the center of the central bright fringe. For what wavelength of light will thefirst-order dark fringe be observed at this same point on thescreen? Express your answer in micrometers(not in nanometers).

λ  = 3.66 × μ m

Explanation:

m=0 ( first order dark fringe), λ = 610 × 10⁻⁹ m, L= 3 m (distance from slit),

y= 4.84 10⁻³ m ( dark fringe distance)

Solution: we have find first the d by equation

y = L λ /d

d= L λ /L = 3 × 610 × 10⁻⁹ m / 4.84 10⁻³ m

d= 3.781 × 10⁻⁴ m

Now by formula for destructive interference for m=0

y = (m +1/2) λ / d

4.84 10⁻³ m = (0 +0.5) λ  / 3.781 × 10⁻⁴ m

λ = (4.84 10⁻³ m × 3.781 × 10⁻⁴ m) / 0.5

λ = 0.000003660008

λ  = 3.66 × 10⁻⁶ m

λ  = 3.66 × μ m