Question

A film of oil on water will appear dark when it is very thin because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the oil can be and appear dark at all visible wavelengths? Oil has an index of refraction of 1.40.

a) 200 nm
b) 225 nm
c) 250 nm
d) 275 nm

Answer 1

The thickest the oil can be and still appear dark at all visible wavelengths is when the path length difference is less than one-fourth the wavelength. By calculating the thickest oil using the provided formula and values, we find that the correct answer is b) 225 nm.

Step-by-step explanation:

The thickest the oil can be and still appear dark at all visible wavelengths is when the path length difference is less than one-fourth the wavelength. To find the thickest oil, we need to consider the conditions for maximum destructive interference. When the path length difference is less than one-fourth the wavelength, the light waves interfere destructively, resulting in darkness.

The formula to calculate the path length difference is:

path length difference = 2 x thickness x refractive index - (wavelength / 4)

If we rearrange the formula to solve for the thickness, we get:

thickness = [(wavelength / 4) - (2 x thickness x refractive index)] / 2

By substituting the values provided: wavelength = 400-700 nm, refractive index = 1.40, and path length difference = wavelength/4, we can calculate the thickest the oil can be.

The thickest oil is approximately 225 nm, so the correct answer is b) 225 nm.