In a Young's double slit experiment, the intensity at the central maximum is I₀. The intensity at a distance β/4 from the central maximum is (β is fringe width)

CONCEPT:

Young's double-slit experiment

• Young’s double-slit experiment helped in understanding the wave nature of light.
• The original Young’s double-slit experiment used diffracted light from a single monochromatic source of light.
• The light that comes from the monochromatic source is passed into two slits to be used as two coherent sources.
• At any point on the screen at a distance ‘y’ from the center, the waves travel distances l1 and l2 to create a path difference of Δl at that point.
• If there is a constructive interference on the point then the bright fringe occurs.
• If there is a destructive interference on the point then the dark fringe occurs.

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Fringe width (β):

• The separation between any two consecutive bright or dark fringe is called fringe width.
• In Young’s double-slit experiment all fringes are of equal width.
• In Young’s double-slit experiment the fringe width is given as,

\(⇒ β=\frac{λ D}{d}\)

Where d = distance between slits, D = distance between slits and screen, and λ = wavelength

CALCULATION:

Given the Intensity of central maximum = I_o, and y = \(\frac{\beta}{4}\)

Let the intensity of each source is I.

Then the intensity of central maxima is given as,

⇒ I₀ = 4I     -----(1)

• In Young’s double-slit experiment the fringe width is given as,

\(⇒ β=\frac{λ D}{d}\)     -----(2)

We know that the path difference at a distance y from the central maxima is given as,

\(⇒ Δ x=\frac{yd}{D}\)

\(⇒ Δ x=\frac{\beta d}{4D}\)     -----(3)

By equation 2 and equation 3,

\(⇒ Δ x=\frac{λ D}{d}\times\frac{ d}{4D}\)

\(⇒ Δ x=\frac{\lambda}{4}\)     -----(4)

• The phase difference is given as,

\(⇒ Δϕ=\Delta x\left(\frac{2\pi}{\lambda}\right)\)     -----(5)

By equation 4 and equation 5,

\(⇒ Δϕ=\frac{\lambda}{4}\times\left(\frac{2\pi}{\lambda}\right)\)

\(⇒ Δϕ=\frac{\pi}{2}\)     -----(6)

We know that the intensity at a distance y is given as,

⇒ l' = 2l (1 + cos Δϕ)     -----(7)

By equation 1 and equation 7,

\(⇒ I'=2\left(\frac{I_0}{4}\right)\left(1+cos\frac{\pi}{2}\right)\)

\(⇒ I'=\frac{I_0}{2}\)

• Hence, option 2 is correct.