33. (a) When are two monochromatic sources of light said to be coherent? Write the conditions for dark and bright fringes in Young’s double slit experiment. Hence, obtain the expression for the
fringe width.
(b) Show that the intensity of the bright fringe is 4a2 and that of the dark fringe is zero. (Here ‘a’ represents the amplitude of the interfering wave.)
Answers
Explanation:
Coherent sources are those which have exactly the same frequency and are in this same phase or have a zero or constant difference.
Conditions:
(i) The sources should be monochromatic and originating from common single source.
(ii) The amplitudes of the waves should be equal.
(a) Condition for formation of bright and dark fringes.
Suppose a narrow slit S is illuminated by monochromatic light of wavelength λ.
The light rays from two coherent sources S1 and S2 are reaching a point P, have a path difference (S2P - S1 P). (i) If maxima (bright fringe) occurs at point P, then S
2
P−S
1
P=nλ (n=0,1,2,3,...)
(ii) If minima (dark fringe) occurs at point P, then s
2
P−S
1
P=(2n+1)
2
λ
(n=0,1,2,3...)
Light waves starting from S and fall on both slits S
1
and S
2
. Then S
1
and S
2
behave like two coherent sources. Spherical waves emanating from S
1
and S
2
superpose on each other, and produces interference pattern on the screen. Consider a point P at a distance x from 0. the centre of screen. The position of maxima (or minima) depends on the path difference. (S
2
T=S
2
P—S
1
P). From right angled ΔS
2
BP and ΔS
1
AP.
(S
2
P)
2
−(S
1
P)
2
=[D
2
+(x+
2
d
)
2
]−[D
2
+(x−
2
d
)
2
]=2xd
(S
2
P+S
1
P)(S
1
P)(S
2
P−S
1
P)=2xd
S
2
P−S
1
P=
2D
2xd
=
D
xd
.......(i)
For constructive interference (Bright fringes)
Path difference,
D
dx
=nλ where n = 1, 2,3 ,...
For n = 0 , x
0
=0 central brigth fringe
For n = 1, x
1
=
d
Dλ
1st bright fringe
For n = 2, x
2
=
d
2Dλ
2nd bright fringe
For n = n, x
n
=
d
nDλ
nth bright fringe
The distance between two consecutive bright fringes is
β=x
n
−x
n−1
β
′
=(2n−1)
2d
Dλ
−2(n−1)−1
2d
Dλ
=
d
Dλ
For destructive interference (dark fringes)
Path difference
D
dx
=(2n−1)
2
λ
x=(2n−1)
2d
Dλ
where n = 1, 2, 3,....
For n = 1, x
1
′
=
2d
Dλ
for1st dark fringe
For n = 2, x
2
′
=
2d
3Dλ
for 2nd dark fringe
For n = n, x
n
′
=(2n−1)
2d
Dλ
for nth dark fringe
The distance between two consecutive dark fringe is
β
′
=(2n−1)
2d
Dλ
−2(n−1)−1
2d
Dλ
=
d
Dλ
The distance between two consecutive bright or dark fringes is called fringe width (w).
∴ Fringe width =
d
Dl
The expression for fringe width is free from n. Hence the width of all fringes of red light are broader than the fringes of blue light.
(b) Intensity of light (using classical theory) is given as
I∝(Width of the slit)
∝(Amplitude)
2
I
min
I
max
=
(a
1
−a
2
)
2
(a
1
+a
2
)
2
=
9
25
⇒
a
1
−a
2
a
1
+a
2
=
3
5
⇒
a
2
a
1
=
1
4
Intensity ratio
I
2
I
1
=
w
2
w
1
=
a
2
2
a
1
2
⇒
I
2
I
1
=(
1
4
)
2
=
1
16
(c) The condition for the interference fringes to be seen is
b
8
<
d
λ
When s is the size of the source and b is the distance of this source from plane of the slit.