Light of wavelength 5800 Å is incident normally on a slit of width 0.20 mm. A diffraction pattern is observed on screen 2 m away from the slit. From the central bright fringe, the distance of the first dark fringe on either side is

Concept:

The path difference is the difference in the distance travelled by two beams when they are scattered in the same direction from different points. It is equal to d sinθ

Constructive interference (Bright Fringe):

d sinθ = nλ

n = 0: Central (bright) fringe

n = 1: First bright fringe

Destructive interference:

\(d\sin \theta = \left( {n + \frac{1}{2}} \right)λ\)

n = 0: First dark fringe (next to the central fringe)

n = 1: second dark fringe

The separation on the screen between the centres of two consecutive bright fringes or two consecutive dark fringes is called fringe width.

The centres of the bright fringes are obtained at y from the central fringe:

\(y = \frac{D}{d}nλ\)

The centres of the dark fringes are obtained at y from the central fringe:

\(y = \frac{D}{d}\left( {n + \frac{1}{2}} \right)λ\)

The fringe width:

\(w = \frac{D}{d}λ\)

D is the separation between the slits and the screen and d is the separation between the slits.

• The central fringe is n = 0.
• The fringe to either side of the central fringe has an order of n = 1 (the first order fringe).
• The order of the next fringe out on either side is n = 2 (the second-order fringe).

Calculation:

λ = 5800 Å = 5800 × 10^-10 m = 5800 × 10^-7 mm

d = 0.2 mm, D = 2 m = 2000 mm

\( y = \frac{D}{d}\left( {n + \frac{1}{2}} \right)λ \)

\(y = \frac{D}{d} \times \frac{1}{2} \times λ \)

\(= \frac{{2000}}{{0.2}} \times \frac{1}{2} \times 5800 \times {10^{ - 7}}\)

\( λ= 2.9mm \)

The distance of the first dark fringe on either side will be 2 times λ , i.e.