soap bubble

[logistic_guy]

If a soap bubble is $\displaystyle 120 \ \text{nm}$ thick, what color will appear at the center when illuminated normally by while light? Assume that $\displaystyle n = 1.34$.

 

[logistic_guy]

White light means that we have a mix of visible light waves in air. Usually those waves to be seen by the eye, their length lies in the interval:

$\displaystyle 400 \ \text{nm} \leq \lambda \leq 700 \ \text{nm}$

We use the wavelength in air formula to solve this problem.

$\displaystyle \lambda = \frac{2tn}{m + \frac{1}{2}}$

where $\displaystyle m = 0,1,2,\cdots$ are interference orders (or fringe orders).

Let us try $\displaystyle m = 0$.

$\displaystyle \lambda_0 = \frac{2(120 \ \text{nm})(1.34)}{0 + \frac{1}{2}} = 643.2 \ \text{nm}$

Let us try $\displaystyle m = 1$.

$\displaystyle \lambda_1 = \frac{2(120 \ \text{nm})(1.34)}{0 + \frac{1}{2}} = 214.4 \ \text{nm}$

No need to check more values because the wavelengths are getting much smaller. Therefore, the only visible wave has the wavelength:

$\displaystyle \lambda = \textcolor{blue}{643.2 \ \text{nm}}$

This wavelength lies in the $\displaystyle \textcolor{red}{\text{red}}$ color spectrum but it is very close to the $\displaystyle \textcolor{orange}{\text{orange}}$ color spectrum.

Therefore, for safety, it is better to say that the color that will appear at the center is $\displaystyle \textcolor{blue}{\text{orange-red}}$.