In: Physics

# What visible wavelengths of light are strongly reflected from a 390-nm-thick soap bubble?

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?

## Solutions

##### Expert Solution

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 $$(690 \mathrm{nm})$$.