Question

The total change in the path length is ∆l = 2 ∆x = mλ for constructive interference.

source reflection

A coherent red light with λ = 650nm reflects off a moveable sample and interferes with itself in the microscope. The observer sees a bright constructive interference spot. As he moves the tray toward him the spot becomes a dark destructive interference. He continues and counts 24 successive bright spots. Calculate how far the surface has moved from the start.

According to the key, the answer is 7.8 um

please show me how to arrive at this answer.

Answer 1

Given,

Wavelength, λ= 650 nm

Now,

Let the distance travelled by the sample be Δx

From the figure,

In the first case,

distance travelled by the light rays, l1 = 2*l

In the second case,

distance travelled by the light rays, l2 = 2*l - 2*Δx

Thus,

Path difference, Δl = l1 - l2 = 2*l - (2*l - 2*Δx)

= 2*Δx

Now,

Path difference between two successive constructive interference or bright spots is given by

=> Path difference = mλ where m = 1,2,3...

Now,

number of bright light observed is 24

Thus,

Path difference = 24*λ = 24 * 650 * 10^-9 m

= 15.6 * 10^-6 m

Since,

path difference = 2*Δx

=> 2*Δx = 15.6 * 10^-6

=> Δx = (15.6 * 10^-6) / 2 = 7.8 * 10^-6 m

= 7.8 µm

Thus, distance travelled by the sample is 7.8 µm