A flake of glass (refractive index 1.5) is placed over one of the opening of a double-slit apparatus. The interference pattern displaced itself through seven successive maxima toward the side where the flake is placed. If wavelength of the light is `lambda = 600 nm`, then the thickness of the flake is

Options

A. 2100 nm
B. 4200 nm
C. 8400 nm
D. none of above

Explanation

To find the thickness of the glass flake placed over one of the openings of a double-slit apparatus, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a double-slit experiment where a glass flake with a refractive index (μ) of 1.5 is placed over one of the slits. The interference pattern shifts by 7 maxima towards the side where the flake is placed.

2. **Identify the Relevant Formula**:
The shift in the interference pattern can be expressed as:
\[ \Delta y = (μ - 1) \frac{t}{d} \]
where:
- \(\Delta y\) is the shift in the position of the maxima,
- \(μ\) is the refractive index of the glass,
- \(t\) is the thickness of the glass flake,
- \(d\) is the distance between the slits.

In this case, since the shift corresponds to 7 maxima, we can express it as:
\[ \Delta y = 7\beta \]
where \(\beta\) is the fringe width.

3. **Calculate the Fringe Width (\(\beta\))**:
The fringe width (\(\beta\)) is given by:
\[ \beta = \frac{\lambda D}{d} \]
where \(D\) is the distance from the slits to the screen and \(\lambda\) is the wavelength of the light used (600 nm in this case).

4. **Substitute the Shift**:
We can now relate the shift to the thickness of the flake:
\[ 7\beta = (μ - 1) t \]
Rearranging gives:
\[ t = \frac{7\beta}{μ - 1} \]

5. **Substitute \(\beta\)**:
Now we substitute \(\beta\) into the equation:
\[ t = \frac{7 \left(\frac{\lambda D}{d}\right)}{μ - 1} \]

6. **Substitute Known Values**:
We know:
- \(\lambda = 600 \text{ nm} = 600 \times 10^{-9} \text{ m}\)
- \(μ = 1.5\)
Therefore:
\[ μ - 1 = 1.5 - 1 = 0.5 \]
Now substituting into the equation for \(t\):
\[ t = \frac{7 \cdot \lambda}{μ - 1} = \frac{7 \cdot 600 \times 10^{-9}}{0.5} \]

7. **Calculate Thickness**:
\[ t = \frac{4200 \times 10^{-9}}{0.5} = 8400 \times 10^{-9} \text{ m} = 8400 \text{ nm} \]

### Final Answer:
The thickness of the glass flake is \(8400 \text{ nm}\).

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