In double slit experiment fringes are obtained using light of wavelength `4800 Å` One slit is covered with a thin glass film of refractive index. `1.4` and another slit is covered by a film of same thickness but refractive index `1.7`. By doing so, the central fringe is shifted to fifth bright fringe in the original pattern. The thickness of glass film is

Options

A. `2xx10^(-3)mm`
B. `4xx10^(-3)mm`
C. `6xx10^(-3)mm`
D. `8xx10^(-3)mm`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Concept of Path Difference
In a double slit experiment, the interference pattern is created due to the path difference between the light waves coming from the two slits. When a thin film is placed in front of one of the slits, it introduces an additional optical path difference due to the refractive index of the film.

### Step 2: Calculate the Optical Path Difference
The optical path difference (OPD) introduced by a thin film of thickness \( t \) and refractive index \( n \) is given by:
\[ \text{OPD} = t(n - 1) \]
For the two slits:
- Slit 1 has a film of refractive index \( n_1 = 1.4 \)
- Slit 2 has a film of refractive index \( n_2 = 1.7 \)

The total optical path difference between the two slits due to the films is:
\[ \text{Total OPD} = t(n_2 - 1) - t(n_1 - 1) = t[(n_2 - 1) - (n_1 - 1)] = t[(1.7 - 1) - (1.4 - 1)] \]
\[ = t[0.7 - 0.4] = t(0.3) \]

### Step 3: Relate OPD to Fringe Shift
The problem states that the central fringe is shifted to the fifth bright fringe. The condition for bright fringes in a double slit experiment is given by:
\[ \text{OPD} = m \lambda \]
where \( m \) is the order of the fringe (in this case, \( m = 5 \)) and \( \lambda \) is the wavelength of the light.

Given that the wavelength \( \lambda = 4800 \, \text{Å} = 4800 \times 10^{-10} \, \text{m} \), we can write:
\[ t(0.3) = 5 \lambda \]

### Step 4: Substitute Values and Solve for Thickness
Substituting the value of \( \lambda \):
\[ t(0.3) = 5 \times (4800 \times 10^{-10}) \]
\[ t(0.3) = 24000 \times 10^{-10} \]
\[ t = \frac{24000 \times 10^{-10}}{0.3} \]
\[ t = 80000 \times 10^{-10} \, \text{m} \]
\[ t = 8000 \, \text{Å} \]

### Final Answer
The thickness of the glass film is \( 8000 \, \text{Å} \).

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