Bright and Dark Fringe Spacing

Relevant equation for the following scenarios: d sin θ = m λ and d sin θ = (m + ½) λ

Scenario: You shoot a laser beam with a wavelength of 400 nm to illuminate a double slit, with a spacing of 0.002 cm, and produce an interference pattern on a screen 75.0 cm away.

Q1) What is the angle θ between the m = 0 fringe and the m = 1 fringe?

Q2) What is the distance y between the m = 0 fringe and the m = 1 fringe?

Q3) What is the distance between the m = 1 bright fringe and the dark fringe between m =1 and m = 2?

Solutions:

Q1) Using equation d sin θ = m λ , we know:d = 0.002 cm = 0.002 x 10-2 mλ = 400 nm = 400 x 10-9 m

Rearranging the equation, θ = arcsin(λ/d) = arcsin(400 x 10-9 m/0.002 x 10-2 m) = 1.15°

Q2) To obtain a value for y, we can use trigonometry.

tan θ = y/D => y = D*tan θ = (75 x 10-2 m)*tan (1.15°) = 0.015 m

Q3) First we have to solve for the angle between the m = 0 fringe and the dark fringe between m = 1 and m = 2, then find the difference between y found in Q2 and the position of the dark fringe.

Using equation d sin θ = (m + ½) λ , we know:d = 0.002 cm = 0.002 x 10-2 mλ = 400 nm = 400 x 10-9 m(m + ½) = 1.5

Rearranging the equation, θ1/2 = arcsin(((m + ½)λ)/d) = arcsin(1.5*400 x 10-9 m/0.002 x 10-2 m) = 1.72°

Position of the dark fringe: y1/2 = D*tan θ1/2 = 0.0225 m

The dark fringe and the adjacent bright fringe are 0.0075m apart.