
 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 = nl
= (2n)l / 2 where n = 0, 1, 2, 3, …
= even multiple of (l_{2})
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)
S_{1}, S_{2} = coherent sources
d = distance of separation between sources
AB = Screen at a distance D from source
OP = Perpendicular bisector of segment S_{1}S_{2},
meeting screen at P.
Q = Point on screen at a distance x from P.
S_{1} M, S_{2}N = Perpendiculars to screen from
sources S_{1} S_{2} = path difference between the two
light waves reaching at point
Q = S_{2}Q  S_{1}Q



In right angled triangle, S_{2}NQ

= S_{2}Q^{2} = S_{2}N^{2}
+ QN^{2}




= D^{2} + (x +

d


2


)^{2}



...(1)



Similarly, in right angled triangle, S_{2}MQ =

S_{1}Q^{2} = S_{1}M^{2}
+ QM^{2}




= D^{2} + (x 

d


2


)^{2}



...(2)




\ S_{2}Q^{2}  S_{1}Q^{2}

= [D^{2} + (x +

d


2


)^{2}]  [D^{2}  (x 

d


2


)^{2}]








= 2xd







\ (S_{2}Q  S_{1}Q) (S_{2}Q + S_{1}Q)
= 2xd

\ S_{2}Q  S_{1}Q =

2xd


(S_{2}Q + S_{1}Q)



D>> x or d, S_{1}Q = S_{2}Q = D

\ S_{2}Q  S_{1}Q =

2xd


2D


=

xd


D




