INTERFERENCE

 

Principle of superposition:-

 This principle states that the resultant displacement of a particle of the medium acted upon by two or more waves simultaneously is the algebraic sum of the displacements of the same particle due to individual wave, in the absence of the other waves.

 Resultant displacement depends on the amplitudes of the individual waves and phase difference between the waves.

Interference:-

 When two or more light waves superimpose, then the modification in the distribution of intensity is called interference. In interference, intensity is redistributed.

Conditions of for bright and dark bands in interference pattern:-

 Let two waves of amplitudes a1 and a2 are represented by 

y1=a1sinωt -------------- (1) and y2=a2sin⁡(ωt+δ) ---------------- (2)

Here is the phase difference between the two waves.

Resultant displacement y=y1+ y2=a1sinωt+a2sin ωt+δ =a1sinωt+a2sinωtcosδ+a2cosωtsinδ=sinωta1+a2cosδ+cosωta2sinδ --------------- (3)

Let a1+a2cosδ=Rcosθ ----------- (4) and a2sinδ=Rsinθ --------------- (5)

∴(3)⇒ y=Rcosθsinωt+cosωtRsinθ=Rsin(ωt+θ) -------- (6)

Here R is the resultant amplitude 

(4)2+(5)2⇒R2=a12+a22cos2δ+2a1a2cosδ+a22sin2δ=a12+a22+2a1a2cosδ⇒

R=a12+a22+2a1a2cosδ ------------- (7)

Since the intensity is directly proportional to square of amplitude I∝a2

∴ I=I1+I2+2I1I2cosδ ---------------- (8)

Case(i) condition for maximum intensity

 If cosδ=+1, then I becomes maximum i.e. phase difference δ=2nπ (or) path difference x=nλ n=0,1,2,3,……., then resultant intensity becomes maximum.

∴ I=I1+I2+2I1I2Imax=I1+I22

Resultant amplitude, R=a1+a2; If a1=a2;R=2a orI=4a2

When the resultant amplitude is the sum of the amplitudes due to two waves the interference is known as constructive interference.

 

Case(ii) condition for minimum intensity

 If cosδ=-1, then I becomes minimum i.e. phase difference δ=(2n+1)π (or) path difference x=(2n+1)2 n=0,1,2,3,……., then resultant intensity becomes minimum.

∴ I=I1+I2-2I1I2Imax=I1-I22

Resultant amplitude, R=a1-a2; If a1=a2;R=0 orI=0

When the resultant amplitude is equal to the difference of two amplitudes the interference is known as destructive interference.