In a double-slit experiment, the slit separation is 1.75 mm, and two coherent wavelengths of light, 425 nm and 510 nm, illuminate the slits. At what angle from the centerline on either side of the central maximum will a bright fringe from one pattern first coincide with a bright fringe from the other pattern

the required angle is 0.0834879⁰

Explanation:

Given  the data in the question;

slit separation; d = 1.75 mm = 1.75 × 10⁻³ m

wavelength λ₁ = 425 nm = 425 × 10⁻⁹ m

wavelength λ₂ 510 nm = 510 × 10⁻⁹ m

Now, we know that, the angle at which a particular bright fringe occurs on either side of the central bright fringe will be;

tanθ = / D = mλ/d

since they both coincides;

tanθ₁ = tanθ₂

m₁λ₁/d = m₂λ₂/d

multiply both sides by d

so,

m₁/m₂ = λ₂/λ₁

we substitute

m₁/m₂ = 510 nm / 425 nm

m₁/m₂ = 510 nm / 425 nm

divide through by 85

m₁/m₂ = 6 / 5

hence m₁ and m₂ are 6 and 5

so, from the previous formula

tanθ₂ = m₂λ₂/d

we substitute

tanθ₂ = [ 5 × ( 510 × 10⁻⁹ m ) ] / 1.75 × 10⁻³ m

tanθ₂ = 255 × 10⁻⁸ m / 1.75 × 10⁻³ m

tanθ₂ = 255 × 10⁻⁸ m / 1.75 × 10⁻³ m

tanθ₂ = 0.00145714

θ₂ = tan⁻¹( 0.00145714 )

θ₂ = 0.0834879⁰

Therefore, the required angle is 0.0834879⁰

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