Question

A thin film of oil \((n=1.50)\) is spread over a puddle of water \((n=1.33) .\) In a region where the film looks red from directly above $(\lambda=630 \mathrm{nm}),$ what is the minimum possible thickness of the film? (tutorial: thin film).

Short answer

Answer: The minimum thickness of the oil film is 210 nm.

Step-by-step solution

Understand thin film interference

Thin film interference occurs when light waves reflect off two surfaces separated by a thin film, causing constructive or destructive interference between the reflected waves. In this case, the film is the oil on top of the water. Constructive interference leads to a brighter color, while destructive interference results in a darker color.

Find the interference condition for a minimum thickness

We will use the following equation for the constructive interference of the thin film: $$2nt = m\lambda_{air}$$ where \(n\) is the refractive index of the oil, \(t\) is the film thickness, \(m\) is an integer representing the order of constructive interference, and \(\lambda_{air}\) is the wavelength of light in air. In our case, \(n=1.50\), \(\lambda_{air}=630\,\text{nm}\), and we want to find the smallest possible \(t\) for constructive interference, so we will consider the case when \(m=1\).

Solve for the minimum thickness of the film

Plug in the given values and solve for \(t\): $$2(1.50)t = 1(630\,\text{nm})$$ $$t = \frac{630\,\text{nm}}{2(1.50)}$$ $$t = 210\,\text{nm}$$ Therefore, the minimum possible thickness of the film for it to appear red when viewed from above is 210 nm.