In: Physics

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?

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 \((690 \mathrm{nm})\).

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