A film of soapy water ( \(n=1.33\) ) on top of a plastic cutting board has a thickness of \(233 \mathrm{nm} .\) What color is most strongly reflected if it is illuminated perpendicular to its surface?

The formula for constructive interference in a thin film is given by: \(2 \cdot t \cdot n = m \cdot \lambda\) Where, - \(t\) is the thickness of the film - \(n\) is the refractive index of the film - \(m\) is an integer representing the order of interference - \(\lambda\) is the wavelength of light for that particular color We want to find the color that is most strongly reflected, which corresponds to the first order of interference (\(m = 1\)).

Using the given thickness and refractive index, we can now calculate the wavelength of the reflected light for the first order of interference. We can write the formula like this: \(\lambda = \frac{2 \cdot t \cdot n}{m}\) Plugging in our given values: \(\lambda = \frac{2 \cdot (233 \mathrm{nm}) \cdot 1.33}{1}\) Calculating this value: \(\lambda \approx 619.86 \mathrm{nm}\)

Now that we have found the wavelength of the most strongly reflected light, we can determine the corresponding color. The wavelength of light is related to the color according to the following approximate ranges: - Violet: \(380 \mathrm{nm} - 450 \mathrm{nm}\) - Blue: \(450 \mathrm{nm} - 495 \mathrm{nm}\) - Green: \(495 \mathrm{nm} - 570 \mathrm{nm}\) - Yellow: \(570 \mathrm{nm} - 590 \mathrm{nm}\) - Orange: \(590 \mathrm{nm} - 620 \mathrm{nm}\) - Red: \(620 \mathrm{nm} - 750 \mathrm{nm}\) Our calculated wavelength, \(619.86 \mathrm{nm}\), falls into the range associated with the transition between orange and red light. Therefore, the color most strongly reflected by the film of soapy water is a shade between orange and red.