MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Interference of Light Page 2

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3.

Conditions for constructive and destructuve interference:

• There is constructive interference at a point if the two light waves arrive at the point in phase.

The phase difference between the two waves is 2np or integral multiples of 2p i.e., 0, 2p, 4p, 6p … etc. i.e., it is an even integral multiple of p.

Phase difference of 2p corresponds to a path difference of l, where l is the wavelength of light.

For constructive interference i.e., for bright points,
Path difference = 0,
l, 2l, 3l, … etc.
Path difference = n
l
= (2n)
l / 2 where n = 0, 1, 2, 3, …
= even multiple of (
l2)

The condition for constructive interference or brightness is that the path difference between the two waves arriving at the point should be an integral multiple of the wavelength or it should be an even integral multiple of l / 2.

There is destructive interference at a point (darkness) if the two light waves arrive at that point in opposite phase
i.e., if the phase difference between the waves is
p, 3p, 5p, … etc.

Phase difference 2p ® Path difference l

Phase difference p ® Path difference

 l 2

\ There is destructive interference if path difference is

 l 2

,3

 l 2

,5

 l 2

,..

(2n - 1)

 l 2

Where n = 1, 2, 3 … etc.

The condition for destructive interference or darkness at a point is that the path difference between the two waves arriving at that point should be an odd integral multiple of half wavelength.

Expression for Path Difference (Analytical Method)

S1, S2 = coherent sources
d = distance of separation between sources
AB = Screen at a distance D from source
OP = Perpendicular bisector of segment S1S2, meeting screen at P.
Q = Point on screen at a distance
x from P.
S1 M, S2N = Perpendiculars to screen from sources S1 S2 = path difference between the two light waves reaching at point
Q = S2Q
- S1Q

-

In right angled triangle, S2NQ

= S2Q2 = S2N2 + QN2

= D2 + (x +

 d 2

)2

...(1)

Similarly, in right angled triangle, S2MQ =

S1Q2 = S1M2 + QM2

= D2 + (x

 d 2

)2

...(2)

\ S2Q2 - S1Q2

= [D2 + (x +

 d 2

)2] - [D2 - (x -

 d 2

)2]

= 2xd

\ (S2Q - S1Q) (S2Q + S1Q) = 2xd

\ S2Q - S1Q =

 2xd (S2Q + S1Q)

D>>
x or d, S1Q = S2Q = D

\ S2Q - S1Q =

 2xd 2D

=

 xd D

Expression for Fringe Width (Band Width)

The distance between two consecutive bright bands or dark bands is called the band width or fringe width.

Let xn, xn+1 = distances of nth and (n + 1)th bright bands from central bright band at P.

For nth bright band

Path difference =

 xnd D

= nl (condition for brightness)

For (n + 1)th bright band

… (3)

Path difference =

 xn+ 1d D

= (n + 1)l

Subtracting (3) from (4),

… (4)

\

 (xn+ 1 - xn)d D

=l

\ xn + 1 - xn =

 lD d

xn+ 1 - xn = X (Band width)

\ X =

 lD d

Similarly, for dark bands

(Band width) (bright band) … (5)

 xm d D

= (2m - 1)

 l 2

&

 xm+ 1 d D

= 2m

 l 2

xm + 1 - xm =

 lD d

\ x=

 lD d

(Band width) (dark band) … (6)

From (5) and (6), the distance between bright bands is the same as distance between dark bands Þ Interference fringes are equally and alternately spaced.

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