Class 11th formula seet.

WAVE OPTICS

NATURE OF LIGHT :
Light is an electromagnetic wave which is sinusoidally varying electric and magnetic fields propagated from
one part to another part. The electric and the magnetic field are given by

E y  E0 sin  kx  wt 

Bz  B0 sin  kx  wt 
1
It propagates as transverse non mechanical wave in a medium at a speed given by V 
 ;
The electric and magnetic fields are related as E0= VB0

REFRACTIVE INDEX OF A MEDIUM :
Refractive index of a medium is defined as
speed of light in vaccum c
  
speed of light in med v
INTERFERENCE :
The modification in the distribution of intensity of light in the region of superposition of two coherent light
waves is called interference. At some points the waves superpose in such a way that the resultant intensity
is greater than the sum of the intensities due to separate waves (constructive interference) while at some
other points intensity is less than the sum of the separate intensities (destructive interference).

YOUNG’ DOUBLE SLIT EXPERIEMENT :
Let the two waves each of angular frequency  from sources S1 and S2 reach the point P. Equations are

given by y1  A1 sin  t  kx  and, y2  A2 sin  t  k  x  x  
So the resultant wave at P by principle of superposition will be

y  y1  y2  A1 sin  t  kx   A2 sin  t  kx    ; where  is initail phase difference

y  A sin  t  kx    where, A2  A12  A22  2 A1 A2 cos 

and,   tan1  A2 sin  /  A1  A2 cos   
Superposition of two waves of equal frequencies propagating almost in the same direction, results in harmonic
wave of same frequency  and wavelength    2 / k  but amplitude A. The intensity of resultant wave

I  K  A12  A22  2 A1 A2 cos  
 

I  I1  I 2  2  
I1I 2 cos 
The resultant intensity at P is not just the sum of intensities due to separate waves (I1 + I2) but different and
depends on phase difference  and the position of the point P.

, CONDITION FOR INTERFERENCE

(a) Intensity will be maximum when :

cos   max .  1
or   2n with n = 0, 1, 2

2
 x   2n

x  n


I max  I1  I 2  2 I1I 2 
2 2
I max   I1  I 2    A1  A2 

Intensity will be maximum at those points where path difference is an integral multiple of wavelength  and
maximum intensity is greater than the sum of two intensities (I1 + I2). These points are called points of
constructive interference or interference maxima.
(b) Intensity will be minimum when :

2
cos   min .  1 ;   , 3, 5 ;     2n  1  ;  x     2n  1  ; x    2n  1  / 2

2
 
I min  I1  I 2  2 I1I 2  I min   I1  I 2    A1  A2 
2


Intensity will be minimum at those points where path difference is an odd integral multiple of   / 2  and
minimum intensity is less than the sum of two intensities (I1 + I2). These points are called points of destructive
interference or interference minima.
2 2
I max   I1  I2  I min   I1 ~ I 2 
2 2
I max  I1  I 2   A1  A2 
I1 A12
  
I min 2 2 ;
 I1  I2   A1  A2  I 2 A22

 All maxima are equally spaced (as path difference between two consecutive maxima is  ) and equally
bright the two waves from S1 and S2 have same frequency and start in the same phase at P they have
a constant phase difference    2 /   x, developed due to different paths traversed by them.
Such waves are said to be ‘Coherent’ and produce sustained interference effects.
If d is the separation between the slits and D (>>d) is the distance of screen from the plane of slits as

x  d sin  sin    x / d 
for small , sin   tan      y / D 
y x D
  x  y
D d d
 If the point P is nth bright fringe, x  n and hence