Question

at a particular spot on an oil film on top of water (n = 1.41), you see purple light at 388 nm. if that is from the second longest (m = 2) possible wavelength, how thick is the oil spot at that point? ? nm

Explanation:

Step1: Recall thin film interference formula

For constructive interference in a thin film where the film has a higher refractive index than the surrounding medium (air, \(n\approx1\)) and the substrate (water, \(n_{water}\approx1.33 < 1.41\)), there is a phase shift of \(\lambda/2\) only at the top surface. The condition for constructive interference is:
$$2nt = m\lambda$$
where \(n=1.41\), \(m=2\), \(\lambda=388\ \text{nm}\), and \(t\) is the film thickness.

Step2: Rearrange for thickness \(t\)

Isolate \(t\) by dividing both sides by \(2n\):
$$t = \frac{m\lambda}{2n}$$

Step3: Substitute values and calculate

Plug in \(m=2\), \(\lambda=388\ \text{nm}\), \(n=1.41\):
$$t = \frac{2\times388}{2\times1.41} = \frac{388}{1.41} \approx 275\ \text{nm}$$

Answer:

275 nm