A portion of a soap bubble appears green ( `lambda = 500.0 nm` in vacuum) when viewed at normal incidence in white light, Determine the two smallest, non-zero thicknesses for the soap film if its index of refraction is 1.40.

Options

A. 89 nm and 179 nm
B. 125 nm and 250 nm
C. 89 nm and 268 nm
D. 125 nm and 375 nm.

Answer

To solve the problem of determining the two smallest non-zero thicknesses for a soap bubble that appears green (λ = 500 nm) when viewed at normal incidence, we can follow these steps:

### Step 1: Understand the condition for constructive interference
For thin films, constructive interference occurs when the path difference between the light waves reflected from the top and bottom surfaces of the film leads to a phase difference that is an integer multiple of the wavelength. The condition for constructive interference at normal incidence is given by:

\[ 2nt = (m + \frac{1}{2})\lambda \]

where:
- \( n \) is the refractive index of the film,
- \( t \) is the thickness of the film,
- \( \lambda \) is the wavelength of light in vacuum,
- \( m \) is an integer (0, 1, 2, ...).

### Step 2: Rearrange the equation for thickness
To find the thickness \( t \), we rearrange the equation:

\[ t = \frac{(m + \frac{1}{2})\lambda}{2n} \]

### Step 3: Substitute the known values
Given:
- \( \lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)
- \( n = 1.40 \)

Substituting these values into the equation for \( t \):

\[ t = \frac{(m + \frac{1}{2}) \times 500 \times 10^{-9}}{2 \times 1.40} \]

### Step 4: Calculate the thickness for the smallest non-zero values of \( m \)
1. For \( m = 0 \):
\[ t_1 = \frac{(0 + \frac{1}{2}) \times 500 \times 10^{-9}}{2 \times 1.40} = \frac{250 \times 10^{-9}}{2.8} = 89.2857 \, \text{nm} \]

2. For \( m = 1 \):
\[ t_2 = \frac{(1 + \frac{1}{2}) \times 500 \times 10^{-9}}{2 \times 1.40} = \frac{(1.5) \times 500 \times 10^{-9}}{2.8} = \frac{750 \times 10^{-9}}{2.8} = 267.8571 \, \text{nm} \]

### Step 5: Finalize the results
Thus, the two smallest non-zero thicknesses for the soap bubble are approximately:
- \( t_1 \approx 89.29 \, \text{nm} \)
- \( t_2 \approx 267.86 \, \text{nm} \)

### Summary of Results
- First thickness: \( t_1 \approx 89.29 \, \text{nm} \)
- Second thickness: \( t_2 \approx 267.86 \, \text{nm} \)