The central fringe of the interference pattern produced by the light of wavelength 6000 Å is found to shift to the position of 4th dark fringe after a glass sheet of refractive index 1.5 is introduced. The thickness of glass sheet would be

Options

A. `4.8 mu m`
B. `4.2 mu m`
C. `5.4 mu m `
D. `3.0 mu m `

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the Shift of the Central Fringe
The central fringe shifts to the position of the 4th dark fringe when a glass sheet is introduced. The path difference for the dark fringes is given by the formula:
\[ \text{Path difference} = \left(n + \frac{1}{2}\right) \lambda \]
where \( n \) is the order of the dark fringe. For the 4th dark fringe, \( n = 3 \):
\[ \text{Path difference} = \left(3 + \frac{1}{2}\right) \lambda = \frac{7}{2} \lambda \]

### Step 2: Calculate the Path Difference Due to the Glass Sheet
When a glass sheet of thickness \( T \) and refractive index \( \mu \) is placed in front of one of the slits, it introduces an additional path difference. The path difference \( \Delta x \) introduced by the glass sheet is given by:
\[ \Delta x = (\mu - 1) T \]
Here, the refractive index \( \mu \) of the glass sheet is 1.5.

### Step 3: Set Up the Equation
We know that the path difference created by the glass sheet must equal the path difference for the 4th dark fringe:
\[ (\mu - 1) T = \frac{7}{2} \lambda \]
Substituting \( \mu = 1.5 \):
\[ (1.5 - 1) T = \frac{7}{2} \lambda \]
This simplifies to:
\[ 0.5 T = \frac{7}{2} \lambda \]

### Step 4: Solve for Thickness \( T \)
Now, we can solve for \( T \):
\[ T = \frac{7 \lambda}{0.5} = 14 \lambda \]

### Step 5: Substitute the Wavelength
The given wavelength \( \lambda \) is 6000 Å (angstroms). We need to convert this to meters for standard SI units:
\[ \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \]
Now substituting this value into the equation for \( T \):
\[ T = 14 \times (6 \times 10^{-7} \, \text{m}) = 8.4 \times 10^{-6} \, \text{m} \]

### Step 6: Convert to Micrometers
To express \( T \) in micrometers:
\[ T = 8.4 \times 10^{-6} \, \text{m} = 8.4 \, \mu m \]

### Final Answer
Thus, the thickness of the glass sheet is:
\[ \boxed{8.4 \, \mu m} \]