Question

Derive an equation for maxima and minima due to interference of light transmitted from thin film of uniform thickness

Answer 1

Consider a thin film of uniform thickness (t) and R.I (μ)

On Reflected side,

The ray of light BF and DE will interfere. The path difference between BF and DE is,

Δ=μ(BC+CD)−BGΔ=μ(BC+CD)−BG

BC=CD=tcosr..........(1)BC=CD=tcosr..........(1)

Now,

BD = (2t) tan r .......(2)

BG = BD sin i

BG = (2t) tan r sin i

BG=2tμsinr(sinrcosr)BG=2tμsinr(sinrcosr) [μ=sinisinr][μ=sinisinr]

BG=2μtsin2rcosr......(3)BG=2μtsin2rcosr......(3)

Substituting (i) and (iii) in Δ :

Δ=μ(tcosr+tcosr)−2μtsin2rcosrΔ=μ(tcosr+tcosr)−2μtsin2rcosr

=2μtcosr(1−sin2r)=2μtcosr(1−sin2r)

Δ=2μtcosrΔ=2μtcosr

This is a geometric path difference. However, there is a phase change of π, as ray BF is reflected from a denser medium. Hence we need to add ±λ2±λ2 to path difference

Δ=2μtcosr±λ2Δ=2μtcosr±λ2

For Destructive Interference:

Δ=nλΔ=nλ

2μtcosr±λ2=nλ2μtcosr±λ2=nλ

2μtcosr=(2n±1)λ2.....(n=0,1,2,...)2μtcosr=(2n±1)λ2.....(n=0,1,2,...)

This is the required expression for constructive Interference or Maxima.

For Destructive interference:

Δ=(2n±1)λ2Δ=(2n±1)λ2

2μtcosr±λ2=nλ2μtcosr±λ2=nλ

2μtcosr=nλ2μtcosr=nλ

This is the required expression for destructive interference.