# The walls of a soap bubble have about the same index of refraction as that of plain water, n =...

## Question:

The walls of a soap bubble have about the same index of refraction as that of plain water, n = 1.33. There is air both inside and outside the bubble.

Part A. What wavelength (in air) of visible light is most strongly reflected from a point on a soap bubble where its wall is 292 nm thick?

Part B. To what color does this correspond?

a green

b) blue

c) red

d) orange

## Interference:

Interference is the phenomenon that occurs when waves of the same frequency meet and superimpose. There are two types of interference which are the constructive and destructive interference. When the interference results to a wave of higher amplitudes than each of the constituent waves, the process is referred to as constructive interference. Otherwise, it is destructive interference.

Part A. Strong reflectance of a certain color is characterized by constructive interference. Constructive interference is attained when the following condition is satisfied:

{eq}\displaystyle 2t = \left( m + \frac{1}{2} \right) \lambda_w {/eq}

where

• {eq}t {/eq} is the thickness of the soap bubble
• {eq}m = 0, 1 , 2, ... {/eq} is the order of reflection
• {eq}\lambda_w {/eq} is the wavelength in water or in the bubble

Now, we solve {eq}\lambda_w {/eq} by plugging in {eq}m = 0 {/eq} and {eq}t = 292\ \rm nm {/eq} to the above equation:

{eq}\displaystyle \begin{align} 2(292\ \rm nm) & = \left( 0 + \frac{1}{2} \right) \lambda_w \\ \lambda_w &= 1168\ \rm nm\\ \end{align} {/eq}

So the wavelength in air would be:

{eq}\displaystyle \lambda_a = n \lambda_w = (1.33)(1168\ \rm nm) =1553.44\ \rm nm {/eq}

This wavelength is outside the range of the visible spectrum. So we discard this value and solve the equation with {eq}m= 1 {/eq} instead:

{eq}\displaystyle \begin{align} 2(292\ \rm nm) & = \left( 1 + \frac{1}{2} \right) \lambda_w \\ \lambda_w &=389.33\ \rm nm\\ \end{align} {/eq}

{eq}\displaystyle \lambda_a = n \lambda_w = (1.33)(389.33\ \rm nm) =518\ \rm nm {/eq}

Thus, the wavelength (in air) of visible light that is most strongly reflected from a point on a soap bubble is 518 nm.

Part B.

The obtained wavelength in Part A is within the range of the green light wavelengths.