The central fringe shifts to the position of fifth bright fringe, if a thin film of refractive index 1.5 is introduced in the path of light of wavelength `5000 Å`. The thickness of the glass plate is

Options

A. `1mum`
B. `5mum`
C. `3mum`
D. `4mum`

Answer

To solve the problem step by step, we will analyze the given information and apply the relevant formulas.

### Step 1: Understand the problem
We know that the central fringe shifts to the position of the fifth bright fringe when a thin film of refractive index \( \mu = 1.5 \) is introduced in the path of light with a wavelength \( \lambda = 5000 \, \text{Å} \).

### Step 2: Identify the formula for fringe shift
The formula for the shift in the position of the fringe due to the introduction of a thin film is given by:
\[ \Delta y = ( \mu - 1 ) \frac{t \lambda D}{d} \]
where:
- \( \Delta y \) is the fringe shift,
- \( \mu \) is the refractive index of the film,
- \( t \) is the thickness of the film,
- \( \lambda \) is the wavelength of light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the separation between the slits.

### Step 3: Set up the equation for the fifth bright fringe
The position of the fifth bright fringe can be expressed as:
\[ y_5 = \frac{5 \lambda D}{d} \]
Since the central fringe shifts to the position of the fifth bright fringe, we can set \( \Delta y = y_5 \).

### Step 4: Equate the two expressions
From the previous steps, we can equate the expressions:
\[ ( \mu - 1 ) \frac{t \lambda D}{d} = \frac{5 \lambda D}{d} \]

### Step 5: Simplify the equation
We can cancel \( \lambda D/d \) from both sides of the equation:
\[ \mu - 1 = 5 \]

### Step 6: Solve for thickness \( t \)
Substituting \( \mu = 1.5 \):
\[ 1.5 - 1 = 5 \implies 0.5 t = 5 \]
Now, solving for \( t \):
\[ t = \frac{5}{0.5} = 10 \]

### Step 7: Substitute the value of \( \lambda \)
Now, substituting \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \):
\[ t = 10 \times 5000 \, \text{Å} = 10 \times 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-6} \, \text{m} \]

### Step 8: Convert to micrometers
Converting \( 5 \times 10^{-6} \, \text{m} \) to micrometers:
\[ t = 5 \, \mu m \]

### Final Answer
The thickness of the glass plate is \( 5 \, \mu m \).

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