Question

Δϕ=2π (2t/λ)+π = π (4t/λ ​ +1) (eq.3)

For destructive interference to occur between the rays, the phase difference must be some odd integer multiple of π, so
Δϕ=(2m+1)π, (eq.4)
where m=0,1,2,… is a positive integer or zero. 3. Using Eqs. (3) and (4), show that for destructive interference to occur, the relationship between the gap thickness t and the light wavelength λ must be 2t=mλ

Answer 1

For destructive interference in a thin film, the thickness of the film 't' must be an integer multiple of half the wavelength of the light 'λ'. This is derived by setting the provided equations for phase difference equal to each other and solving for 't'.

Step-by-step explanation:

To find the relationship between the gap thickness t and the light wavelength λ for destructive interference, we start with the given equation (3) Δφ = 2π (2t/λ) + π = π (4t/λ + 1). For destructive interference, equation (4) states that Δφ = (2m+1)π, where m is a non-negative integer. Setting these two equations equal to each other gives us:

π (4t/λ + 1) = (2m+1)π

We can cancel the π from both sides and solve for t:

4t/λ + 1 = 2m + 1

4t/λ = 2m

t = mλ / 2

This shows that for destructive interference to occur in thin film interference scenarios, the thickness of the film (t) must be an integer multiple of half the wavelength of the light (λ).