In a double-slit experiment, fringes are produced using light of wavelength `4800A^(@)`. One slit is covered by a thin plate of glass of refractive index 1.4 and the other slit by another plate of glass of double thickness and of refractive index 1.7. On doing so, the central bright fringe shifts to a position originally occupied by the fifth bright fringe from the center. find the thickness of the glass plates.

Options

A. `2.4mum`
B. `4.8 mum`
C. `8mum`
D. `16mum`

Explanation

To solve the problem step by step, we will follow the concepts of wave optics and the double-slit experiment.

### Step 1: Understand the Problem
In a double-slit experiment, we have two slits. One slit is covered with a thin plate of glass of refractive index \( \mu_1 = 1.4 \) and the other slit is covered with a plate of glass of double thickness and refractive index \( \mu_2 = 1.7 \). The central bright fringe shifts to the position of the fifth bright fringe.

### Step 2: Define the Wavelength
The wavelength of the light used is given as \( \lambda = 4800 \, \text{Å} = 4800 \times 10^{-10} \, \text{m} \).

### Step 3: Write the Equation for the Position of Bright Fringes
The position of the \( n \)-th bright fringe in a double-slit experiment is given by:
\[ x_n = \frac{n \lambda D}{d} \]
where \( D \) is the distance from the slits to the screen and \( d \) is the distance between the slits.

For the fifth bright fringe (\( n = 5 \)):
\[ x_5 = \frac{5 \lambda D}{d} \]

### Step 4: Calculate the Phase Change Due to the Glass Plates
The phase change introduced by the glass plates can be expressed as:
\[ \Delta x = ( \mu_2 - \mu_1 ) t \frac{D}{d} \]
where \( t \) is the thickness of the glass plate covering the second slit.

### Step 5: Set Up the Equation for the Shift
Since the central bright fringe shifts to the position of the fifth bright fringe, we can equate the two expressions:
\[ \frac{5 \lambda D}{d} = ( \mu_2 - \mu_1 ) t \frac{D}{d} \]

### Step 6: Simplify the Equation
Cancel \( D \) and \( d \) from both sides:
\[ 5 \lambda = (\mu_2 - \mu_1) t \]

### Step 7: Substitute the Values
Substituting the values of \( \mu_1 \) and \( \mu_2 \):
\[ \mu_2 - \mu_1 = 1.7 - 1.4 = 0.3 \]
Thus, the equation becomes:
\[ 5 \lambda = 0.3 t \]

### Step 8: Solve for Thickness \( t \)
Rearranging the equation gives:
\[ t = \frac{5 \lambda}{0.3} \]
Substituting \( \lambda = 4800 \times 10^{-10} \, \text{m} \):
\[ t = \frac{5 \times 4800 \times 10^{-10}}{0.3} \]
Calculating this:
\[ t = \frac{24000 \times 10^{-10}}{0.3} = 80000 \times 10^{-10} \, \text{m} = 8 \times 10^{-6} \, \text{m} \]

### Step 9: Convert to Microns
Since \( 1 \, \text{m} = 10^6 \, \mu m \):
\[ t = 8 \, \mu m \]

### Final Answer
The thickness of the glass plates is \( 8 \, \mu m \).

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