The smallest thickness of a soap film that appears black when illuminated with 480-nm light is approximately 120 nm.
The phenomenon of thin film interference can be utilized to determine the smallest thickness of a soap film that appears black when illuminated with 480-nm light. For constructive interference, the condition is given by 2t=mλ/n, where t is the thickness of the film, λ is the wavelength of light, m is an integer (in this case, we consider m=1), and n is the refractive index of the medium (air, in this scenario).
Rearranging the formula for t, we get t=λ/2n. Given that the wavelength (λ) is 480 nm and air has a refractive index close to 1, the thickness of the soap film for black appearance is approximately 120nm. At this thickness, destructive interference occurs for the specified wavelength, resulting in the film appearing black when illuminated with 480-nm light.