Coherent Sources

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Conditions for Constructive and Destructive Interference

(Analytical Treatment)

In different colors, the fringes have different widths. This indicates that a relation exists between colour and fringe widths. But before going to the relation, let us know more about the condition under which constructive interference or destructive interference occurs.

If y₁ = A Sin wt and y₂ = B Sin ( wt + f) represent the waves of two coherent sources with A and B as their respective amplitude and f the constant phase difference between the waves.

Then from superposition principle,

y = y₁ + y₂

= A sin wt + B sin( wt + f )

= sin wt( A+ B cos f) + cos wt B sin f

y = R sin (wt + f)

(On expanding the second term)

i.e. put a + cos f = R cos f

Since intensity I a (amplitude) ²

I f (a² + b² + 2ab cosf)

For constructive interference, intensity (I) is maximum when

cosf = +1 or f = 0, 2 p, 4p

f = 2n p where n=0,1,2

Now a path difference of l corresponds to the phase difference 2p. If x is the path difference, then

(Where f is the phase difference)

x = nl

For destructive inference, intensity should be minimum.

cos f = -1

f = p,3 p,5 pp…..

f = (2n - 1)p where n=1,2

Note:

1. During constructive interference

R_max = A + B as cos f = + 1

During destructive inference

cos f = -1

R_min = A - B

special case

If A =B

(i.e., if the two waves have same amplitude) then

This indicates a good contrast between the bright and dark bands, which is required to view a good interference pattern.2. In terms of respective intensities

I₁, I₂ are intensities of waves coming from A and B respectively.

In addition if I₁ = I₂ Then3. If W₁ and W₂ are the widths of two slits from which intensities of light I₁ and I₂ emanate, then

4. Conservation of energy in interference

If there were no interference, intensity of light from two sources at energy point on the screen would be

I = I₁ + I₂ = (a² + b²).

The average intensity of light in the interference pattern is

On comparing the above expression with

This indicates that light energy is redistributed, i.e., from the energy of destructive interference to the regions of constructive interference. No energy is created or destroyed in this process and therefore this phenomenon does not violate the principle of energy conservation.