In a Young's double slit experiment the intensity at the cetral maxima is I₀. Find the intensity at a distance

B(beta)/4 from the central maxima where B(beta) is the fringe width.

Here,

Central maximum intensity = I_o

β = λD/d

y = λD/4d

Now, in Young’s double slit experiment the path difference is given by

∆x = yd/D

=>  ∆x = (λD/4d)d/D = λ/4

Now phase difference corresponding to path difference λ/4

Will be φ = 2π/λ(λ/4) = π/2

Intensity due to interference is given by

I = I₁ + I₂ +2√(I₁I₂)cosφ

At central maximum φ = 0, I = I_o, Let intensity of light from each slit I₁ = I₂ = I_s

 => I₀ = I₁ + I₂ +2√(I₁I₂)

=> I_o = 4I_s ---1.  

Now intensity at a distance β/4 is given by, cosφ = π/2

I = I_s + I_s +2√(I₁I₂)cos π/2

=> I = 2I_s ---2

By 1 and 2.

I = I_o/2

Thus intensity at β/4 is I₀/2