A plane wave of monochromatic light falls normally on a uniform thin or oil which covers a glass plate. The wavelength of source can be varied continuously. Complete destructive is observed for `lambda = 5000 Å` and `lambda = 1000 Å` and for no other wavelength in between. If `mu` of oil is 1.3 and that of glass is 1.5, the thickness of the film will be

Options

A. `6.738 xx 10(-5) cm`
B. `5.7 xx 10^(-5) cm`
C. `4 xx 10^(-5) cm`
D. `2.8 xx 10^(-5) cm`

Explanation

To solve the problem step by step, we will analyze the conditions for destructive interference in a thin film of oil over a glass plate.

### Step 1: Understanding the Setup
We have a thin film of oil (refractive index \( \mu = 1.3 \)) on a glass plate (refractive index \( n = 1.5 \)). Light is incident normally on the film, and we observe complete destructive interference at two specific wavelengths: \( \lambda_1 = 5000 \, \text{Å} \) and \( \lambda_2 = 1000 \, \text{Å} \).

### Step 2: Path Difference Condition for Destructive Interference
For destructive interference in a thin film, the condition is given by:
\[ \Delta x = 2 \mu t = (m + \frac{1}{2}) \lambda \]
where \( m \) is an integer (0, 1, 2, ...), \( \lambda \) is the wavelength of light, and \( t \) is the thickness of the film.

### Step 3: Setting Up Equations for Both Wavelengths
For the two wavelengths, we can write:
1. For \( \lambda_1 = 5000 \, \text{Å} \):
\[ 2 \mu t = (m_1 + \frac{1}{2}) \lambda_1 \]
2. For \( \lambda_2 = 1000 \, \text{Å} \):
\[ 2 \mu t = (m_2 + \frac{1}{2}) \lambda_2 \]

### Step 4: Equating the Two Equations
Since both expressions equal \( 2 \mu t \), we can set them equal to each other:
\[ (m_1 + \frac{1}{2}) \lambda_1 = (m_2 + \frac{1}{2}) \lambda_2 \]

### Step 5: Substituting Values
Substituting \( \lambda_1 = 5000 \, \text{Å} \) and \( \lambda_2 = 1000 \, \text{Å} \):
\[ (m_1 + \frac{1}{2}) \cdot 5000 = (m_2 + \frac{1}{2}) \cdot 1000 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 5000 m_1 + 2500 = 1000 m_2 + 500 \]
\[ 5000 m_1 - 1000 m_2 = -2000 \]

### Step 7: Solving for Integer Values
We can express \( m_2 \) in terms of \( m_1 \):
\[ m_2 = 5 m_1 + 2 \]

### Step 8: Substituting Back to Find Thickness
Now, we can substitute \( m_1 \) and \( m_2 \) back into either of the original equations to find \( t \). Using \( m_1 = 0 \) (the first order):
\[ 2 \cdot 1.3 \cdot t = (0 + \frac{1}{2}) \cdot 5000 \]
\[ 2.6 t = 2500 \]
\[ t = \frac{2500}{2.6} \approx 961.54 \, \text{Å} \]

### Final Answer
The thickness of the oil film is approximately \( t \approx 961.54 \, \text{Å} \).

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