Interference

Principle of superposition: When two or more waves travel through a same medium simultaneously, the resultant displacement at any point is the vector sum of displacement due to the individual waves.

i.e. $\vec{y}=\overrightarrow{y_1}+\overrightarrow{y_2}+\overrightarrow{y_3}+...............$

Coherent sources are two sources of light, which continuously emits light waves of same wavelength or frequency and always are in same phase or have a constant phase difference.

The phenomenon of such redistribution of energy in the resultant light wave formed by the superposition of two light waves having same frequency or wavelength and constant phase difference is called interference of light.

When the crest of one wave overlaps with the crest of another or trough of one wave overlaps the trough of another, their individual effects add up together. The resultant is a wave of increased amplitude. This is known as constructive interference.

If the crest of one wave overlaps with the trough of another, their individual effects are reduced. The resultant is the wave of decreased amplitude. This is known as destructive interference.

Optical path is the product of the distance travelled by the light in the medium and the refractive index of that medium.

Useful Information

• Coherent sources produced by either of division of amplitude or division of wavelength.
• Two independent sources can never be coherent sources.
• Soap bubble is seen colorful due to interference.
• White light gives coloured fringes.
• In the Young's double slit centre is bright and in Newton's ring centre is dark.
• Interference follows the law of conservation of energy.
• If the glass plate is kept in one of the slit, then the breadth of the fringes does not changes.
• An excessively thin film appears black in reflected light.

Key formula

1. Relation between path difference and phase difference : $\varphi$=$\frac{2\pi }{\lambda }$$\times x$
2. Optical path: L=$\mu .d$ where L is distance travelled in free space and d is distance travelled in medium for light wave.
3. $\frac{I_1}{I_2}$=$\frac{a_1^2}{a_2^2}$, where $I_1$and $I_2$are the intensities and $a_1$, $a_2$ are the amplitudes of the waves.
4. R=$\sqrt{a_1^2+a_2^2+2a_1{a}_{2}cos\varphi }$ , R is the amplitude of the resultant wave.
5. Breadth of the fringe: ${\beta }_a$=$\frac{{\lambda }_a.D}{d}$ ( in air )
6. ${\beta }_m$=$\frac{{\lambda }_m.D}{d}$=$\frac{{\lambda }_a.D}{\mu .d}$=$\frac{{\beta }_a}{\mu }$( in the medium)
7. Condition for maxima: $\varphi$=2n$\pi$(n=0,1,2........) ; x=n$\lambda$
8. Condition for minimum: $\varphi$=(2n+1)$\pi$,n=0,1,2,....... ; $\varphi$=(2n-1)$\pi$, n=0,1,2,3....... ; X=(2n$\pm$1)$\frac{\lambda }{2}$