Question

(i) In Young's double-slit experiment, deduce the condition for (a) constructive and (b) destructive interference at a point on the screen. Draw a graph showing variation of intensity in the interference pattern against position 'x' on the screen.

(b) Compare the interference pattern observed in Young's double-slit experiment with single-slit diffraction pattern, pointing out three distinguishing features.

Solution

Expression for fringe width in Young's Double Slit Experiment

Let S₁ and S₂ be two slits separated by a distance d.
GG' is the screen at a distance D from the slits S₁ and S₂.
Point C is equidistant from both of the slits.
The intensity of light will be maximum at this point because the path difference of the waves reaching this point will be zero.
At point P, the path difference between the rays coming from the slits is given by,
S₁ = S₂ P - S₁ P
Now, S₁ S₂ = d, EF = d, and S₂ F = D
In $increment straight S subscript 2 PF$
S₂P = $S subscript 2 P space equals space square root of straight S subscript 2 straight F squared plus PF squared end root space S subscript 2 P italic space italic equals italic space open square brackets D to the power of italic 2 italic space italic plus italic space open parentheses x italic plus begin display style d over italic 2 end style close parentheses to the power of italic 2 over D to the power of italic 2 close square brackets to the power of begin inline style bevelled italic 1 over italic 2 end style end exponent italic space italic space italic space italic space italic space italic space italic space italic space italic space italic equals italic space D italic space open square brackets italic 1 italic plus open parentheses x italic plus begin display style d over italic 2 end style close parentheses to the power of italic 2 over D to the power of italic 2 close square brackets to the power of begin inline style bevelled italic 1 over italic 2 end style end exponent S i m i l a r l y italic comma italic space i n italic space italic increment S subscript italic 1 P E italic comma S subscript italic 1 P italic space italic equals italic space D italic space open square brackets italic 1 italic plus italic 1 over italic 2 open parentheses x italic plus d over italic 2 close parentheses to the power of italic 2 over D to the power of italic 2 close square brackets italic space italic minus italic space D open square brackets italic 1 italic plus italic 1 over italic 2 open parentheses x italic minus d over italic 2 close parentheses to the power of italic 2 over D to the power of italic 2 close square brackets O n italic space e x p a n d i n g italic space b i n o m i a l l y italic comma italic space w e italic space g e t S subscript italic 2 P italic space italic minus italic space S subscript italic 1 P italic space italic equals italic space fraction numerator italic 1 over denominator italic 2 D end fraction open square brackets italic 4 x d over italic 2 close square brackets italic space italic equals italic space fraction numerator x d over denominator D end fraction$
For constructive interference, the path difference is an integral multiple of wavelengths, that is, path difference is n $straight lambda$ .
$Therefore comma space nλ space equals space xd over straight D straight x space equals space nλD over straight d semicolon space where space straight n space equals space 0 comma 1 comma 2 comma 3 comma 4 comma.... Similarly space for space destructive space interference comma space straight x subscript straight n space equals space left parenthesis 2 straight n minus 1 right parenthesis space straight lambda over 2 straight D over straight d$
Graph of intensity distribution in young's double slit experiment is,

ii)
Three distinguishing features observed in Young's Double Slit experiment as compared to single slit diffraction pattern is,
1. In the interference pattern, all the bright fringes have the same intensity. The bright fringes are not of the same intensity in a diffraction pattern.
2. In interference pattern, the dark fringes have zero or small intensity so that the bright and dark fringes can be easily distinguished. While in diffraction pattern, all the dark fringes are not of zero intensity.
3. In interference pattern, the width of all fringes are almost the same, whereas in diffraction pattern, the fringes are of different widths.

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