A certain region of a soap bubble reflects red light of vacuum wavelength `lambda = 650nm`. What is the minimum thickness that this region of the soap bubble could be have? Take the index of reflection of the soap film to be 1.41.

Options

A. `1.2 xx 10^(-7) m`
B. `650 xx 10^(-9) m`
C. `120 xx 10^(7) m`
D. `650 xx 10^(5) m`

Explanation

To find the minimum thickness of the soap bubble that reflects red light of wavelength \( \lambda = 650 \, \text{nm} \) (or \( 650 \times 10^{-9} \, \text{m} \)), we can use the concept of interference in thin films. Here's the step-by-step solution:

### Step 1: Understanding the Interference Condition
When light reflects off a thin film (like a soap bubble), there are two reflections to consider:
1. The first reflection occurs at the top surface (air-soap interface).
2. The second reflection occurs at the bottom surface (soap-air interface).

For constructive interference, the optical path difference must be an integer multiple of the wavelength. However, since there is a phase change of \( \lambda/2 \) upon reflection at the denser medium (soap), we need to account for this in our calculations.

### Step 2: Setting Up the Optical Path Difference
The optical path difference (OPD) for the light reflecting off the two surfaces is given by:
\[ \text{OPD} = 2 \mu t - \frac{\lambda}{2} \]
where:
- \( \mu \) is the refractive index of the soap film (given as \( 1.41 \)),
- \( t \) is the thickness of the soap film,
- \( \lambda \) is the wavelength of light in vacuum.

For constructive interference, we set the OPD equal to \( m \lambda \) (where \( m \) is an integer):
\[ 2 \mu t - \frac{\lambda}{2} = m \lambda \]

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ 2 \mu t = m \lambda + \frac{\lambda}{2} \]
\[ 2 \mu t = (m + \frac{1}{2}) \lambda \]
\[ t = \frac{(m + \frac{1}{2}) \lambda}{2 \mu} \]

### Step 4: Finding Minimum Thickness
To find the minimum thickness, we take \( m = 0 \):
\[ t_{\text{min}} = \frac{(0 + \frac{1}{2}) \lambda}{2 \mu} = \frac{\lambda}{4 \mu} \]

### Step 5: Substituting Values
Now, substituting the values of \( \lambda \) and \( \mu \):
\[ t_{\text{min}} = \frac{650 \times 10^{-9} \, \text{m}}{4 \times 1.41} \]
Calculating this gives:
\[ t_{\text{min}} = \frac{650 \times 10^{-9}}{5.64} \approx 115.4 \times 10^{-9} \, \text{m} \approx 1.154 \times 10^{-7} \, \text{m} \]

### Final Answer
Thus, the minimum thickness \( t_{\text{min}} \) of the soap bubble is approximately:
\[ t_{\text{min}} \approx 1.15 \times 10^{-7} \, \text{m} \text{ or } 115.4 \, \text{nm} \]