How Does Soap Film Thickness Affect Color in Thin-Film Interference?

[aaronb]

Homework Statement

A soap film with different thicknesses at different placers has an unknown refractive index n and air on both sides. In reflected light it looks multicolored. One region looks yellow because destructive interference has removed blue (($$\lambda_{vacuum}$$=469nm)) from the reflected light, while anotyher loks magenta because destructive interference has removed green (($$\lambda_{vacuum}$$=555nm)). In these regions the film has the minimum thickness t required for the destructive interference to occur. Find the ration t$$_{magenta}$$/t$$_{yellow}$$

Homework Equations

$$\lambda_{film}$$ = $$\lambda_{vacuum}$$/ n

The Attempt at a Solution

i know from the problem that m =1 because it says minimum thickness. But after that I do not know how I should set up the problem using algebra or physics.

 

[anathema]

Hello! Thank you for your post. Here is a possible solution to the problem:

First, we can use the equation \lambda_{film} = \lambda_{vacuum}/ n to find the refractive index n of the soap film. We know that in the yellow region, \lambda_{vacuum} = 469nm, and in the magenta region, \lambda_{vacuum} = 555nm. We also know that in both regions, m = 1. Plugging these values into the equation, we get:

n_{yellow} = \lambda_{vacuum} / \lambda_{film, yellow} = 469nm / 469nm = 1

n_{magenta} = \lambda_{vacuum} / \lambda_{film, magenta} = 555nm / 555nm = 1

So, we can see that the refractive index of the soap film is the same in both regions, which makes sense since it is the same film.

Next, we can use the equation for destructive interference to find the ratio of the thicknesses t_{magenta} / t_{yellow}. The equation for destructive interference is:

2nt = (m + 1/2) \lambda_{vacuum}

Where n is the refractive index, t is the thickness of the film, m is the order of the interference, and \lambda_{vacuum} is the wavelength of the incident light.

Since we know that in both regions, m = 1, and we just found that n_{magenta} = n_{yellow} = 1, we can set up the following equation:

2t_{magenta} = (1 + 1/2) \lambda_{vacuum} = (3/2) \lambda_{vacuum}

2t_{yellow} = (1 + 1/2) \lambda_{vacuum} = (3/2) \lambda_{vacuum}

Dividing these two equations, we get:

t_{magenta} / t_{yellow} = (3/2) \lambda_{vacuum} / (3/2) \lambda_{vacuum} = 1

Therefore, the ratio of the thicknesses t_{magenta} / t_{yellow} is 1, meaning that the thickness of the film in the magenta region is the same as the thickness in the yellow region. This makes sense