Question

A soap bubble is essentially a thin film of water surrounded by air. The colors you see in soap bubbles are produced by interference. What visible wavelengths of light are strongly reflected from a 390-nm-thick soap bubble? What color would such a soap bubble appear to be?

Answer 1

For $m=0$, the wavelength

$$\lambda=\frac{4 n t}{1}$$

Substitute 1.33 for $n$ and $390 \mathrm{nm}$ for $t$

$$\begin{aligned} \lambda &=\frac{4(1.33)\left(390 \mathrm{nm}\left(\frac{1 \mathrm{~m}}{10^{9} \mathrm{nm}}\right)\right)}{1} \\ &=2074.8 \mathrm{nm} \end{aligned}$$

For $m=1,$ the visible wavelength

$$\begin{aligned} \lambda &=\frac{4(1.33)\left(390 \mathrm{nm}\left(\frac{1 \mathrm{~m}}{10^{9} \mathrm{nm}}\right)\right)}{3} \\ & \approx 690 \mathrm{nm} \end{aligned}$$

For $m=2,$ the visible wavelength

$$\begin{aligned} \lambda &=\frac{4(1.33)\left(390 \mathrm{nm}\left(\frac{1 \mathrm{~m}}{10^{9} \mathrm{nm}}\right)\right)}{5} \\ &=414.96 \mathrm{nm} \\ & \approx 410 \mathrm{nm} \end{aligned}$$

For $m=3,$ the wavelength

$$\begin{aligned} \lambda &=\frac{4(1.33)\left(390 \mathrm{nm}\left(\frac{1 \mathrm{~m}}{10^{9} \mathrm{nm}}\right)\right)}{7} \\ &=296.4 \mathrm{nm} \end{aligned}$$

Values of $m$ greater than 3 lead to values of $\lambda$ that are smaller than $296.4 \mathrm{nm} .$ Thus the wavelengths in the visible spectrum $(400-700 \mathrm{nm})$ that are strongly reflected by the film are $410 \mathrm{nm}$ and $690 \mathrm{nm}$

Hence, the soap bubble would appear purple due to the combination of deep violet $(410 \mathrm{nm})$ and red $(690 \mathrm{nm})$.

the soap bubble would appear purple due to the combination of deep violet $(410 \mathrm{nm})$ and red $(690 \mathrm{nm})$.