What is the minimum thickness of a soap bubble needed for constructive interference in reflected light, if the light incident on the film is 750 nm? Assume that the refractive index for the film is `n=1.33`

Options

A. 282 nm
B. 70.5 nm
C. 141 nm
D. 387 nm

Explanation

To find the minimum thickness of a soap bubble needed for constructive interference in reflected light, we can follow these steps:

### Step 1: Understand the Condition for Constructive Interference
For constructive interference in a thin film, the condition is given by:
\[ 2T = m \lambda' + \frac{\lambda'}{2} \]
where:
- \( T \) is the thickness of the film,
- \( m \) is the order of interference (an integer),
- \( \lambda' \) is the wavelength of light in the medium.

### Step 2: Calculate the Wavelength in the Medium
The wavelength of light in the medium can be calculated using the formula:
\[ \lambda' = \frac{\lambda}{n} \]
where:
- \( \lambda \) is the wavelength of light in vacuum (750 nm),
- \( n \) is the refractive index of the medium (1.33).

Substituting the values:
\[ \lambda' = \frac{750 \, \text{nm}}{1.33} \]

### Step 3: Substitute \( m \) for Minimum Thickness
For minimum thickness, we take \( m = 0 \):
\[ 2T = 0 \cdot \lambda' + \frac{\lambda'}{2} \]
This simplifies to:
\[ 2T = \frac{\lambda'}{2} \]

### Step 4: Solve for Thickness \( T \)
Rearranging the equation gives:
\[ T = \frac{\lambda'}{4} \]

### Step 5: Substitute \( \lambda' \) into the Thickness Formula
Now we can substitute \( \lambda' \) into the thickness formula:
\[ T = \frac{1}{4} \left( \frac{750 \, \text{nm}}{1.33} \right) \]

### Step 6: Calculate the Numerical Value
Calculating \( \lambda' \):
\[ \lambda' = \frac{750 \times 10^{-9} \, \text{m}}{1.33} \approx 563.91 \times 10^{-9} \, \text{m} \]

Now substituting this into the thickness formula:
\[ T = \frac{563.91 \times 10^{-9}}{4} \approx 140.98 \times 10^{-9} \, \text{m} \]
or approximately:
\[ T \approx 141 \, \text{nm} \]

### Final Answer
The minimum thickness of the soap bubble needed for constructive interference in reflected light is approximately **141 nm**.

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