**CBSE NOTES CLASS 12 PHYSICS **

**CHAPTER 10 WAVE OPTICS**

**Interference **

Superposition of two waves, travelling through the same medium, is called interference.

**Constructive interference** occurs when the two interfering waves have a displacement in the same direction and the amplitude of the resulting wave is the addition of amplitudes of interfering waves.

**Destructive interference** occurs when the two interfering waves have a displacement in different direction (out of phase) and the amplitude of the resulting wave is the difference of amplitudes of interfering waves.

Consider two coherent sources S_{1} and S_{2 }a point P for which S_{1}P = S_{2}P. Since the distances S_{1}P and S_{2}P are equal, waves from S_{1 }and S_{2} will take the same time to travel to the point P and waves that originate from S_{1} and S_{2} in phase will also arrive, at the point P, in phase.

If the displacement produced by the source S_{1} at the point P is given by

y_{1} = a cos ωt

then, the displacement produced by the source S_{2} (at the point P) will also be given by

y_{2} = a cos ωt

Thus, the resultant of displacement at P would be given by

y = y_{1} + y_{2} = 2 a cos ωt

Since the intensity is the proportional to the square of the amplitude, the resultant intensity will be given by,

I = 4I_{o}

where I_{o} represents the intensity produced by each one of the individual sources; I_{o} is proportional to a^{2}.

In fact for all points where the phase difference is 2nπ or path difference is nλ, the interference will be constructive.

If the phase difference is $(2\mathrm{n}+1)$π or path difference is $\left(\frac{2\mathrm{n}+1}{2}\right)$λ, the interference will be destructive.

For any other arbitrary point let the phase difference between the two displacements be φ.

y_{1} = a cos ωt ⇒ y_{2} = a cos (ωt + φ)

The resultant displacement will be given by

y = y_{1} + y_{2} = a [cos ωt + cos (ωt + φ]

$$=\mathrm{}2\mathrm{a}\mathrm{cos}\left(\frac{\phi}{2}\right)\mathrm{cos}\left(\mathrm{\omega}\mathrm{t}\mathrm{}+\frac{\phi}{2}\right)$$

Since φ is constant, the amplitude of the resultant displacement is 2a cos $\frac{\phi}{2}$.

Or the intensity I = 4I_{o} cos^{2 }$\frac{\phi}{2}$

If the two sources are not coherent the average intensity will be given by

<I> = 4I_{o} <cos^{2} $\frac{\phi}{2}$>

where angular brackets represent time averaging.

The function cos^{2 }$\frac{\phi}{2}$ will randomly vary between 0 and 1 and the average value will be $\frac{1}{2}$.

The resultant intensity will be given by,

I = 2I_{o}