One slit of a double slits experiment is covered by a thin glass plate of refractive index`1.4`and the other by a thin glass plate of refractive index`1.7` .The point on the screen ,where central bright fringe was formed before the introduction of the glass sheets,is now occupied by the `15th`bright fringe.Assuming that both the glass plates have same thickness and wavelength of light used in `4800Å` ,find the their thickness.

Explanation

To solve the problem, we need to find the thickness of the glass plates that are placed over the slits in the double-slit experiment. Let's break down the solution step by step.

### Step 1: Understanding the Path Difference
In a double-slit experiment, the path difference between the two waves arriving at a point on the screen determines the position of the bright and dark fringes. For the 15th bright fringe, the path difference (\( \Delta x \)) is given by:
\[ \Delta x = m \lambda \]
where \( m \) is the order of the fringe (in this case, \( m = 15 \)) and \( \lambda \) is the wavelength of light used.

### Step 2: Calculate the Path Difference
Given that the wavelength \( \lambda = 4800 \, \text{Å} = 4800 \times 10^{-10} \, \text{m} \), we can calculate the path difference for the 15th bright fringe:
\[ \Delta x = 15 \times 4800 \times 10^{-10} \, \text{m} = 72000 \times 10^{-10} \, \text{m} = 7.2 \times 10^{-6} \, \text{m} \]

### Step 3: Optical Path Length in Glass Plates
When light passes through a medium with a refractive index \( \mu \), the optical path length is given by:
\[ \text{Optical Path Length} = \mu \cdot t \]
where \( t \) is the thickness of the glass plate.

For the two glass plates, we have:
- For the first glass plate (refractive index \( \mu_1 = 1.4 \)):
\[ \text{Optical Path Length}_1 = (1.4) \cdot t \]
- For the second glass plate (refractive index \( \mu_2 = 1.7 \)):
\[ \text{Optical Path Length}_2 = (1.7) \cdot t \]

### Step 4: Calculate the Effective Path Difference
The effective path difference due to the glass plates is given by:
\[ \Delta x = \text{Optical Path Length}_2 - \text{Optical Path Length}_1 \]
Substituting the expressions from above:
\[ \Delta x = (1.7 t) - (1.4 t) = (1.7 - 1.4) t = 0.3 t \]

### Step 5: Equate the Path Difference
Now we can set the calculated path difference equal to the expression we found earlier:
\[ 0.3 t = 7.2 \times 10^{-6} \, \text{m} \]

### Step 6: Solve for Thickness \( t \)
Now, we can solve for \( t \):
\[ t = \frac{7.2 \times 10^{-6}}{0.3} = 2.4 \times 10^{-5} \, \text{m} = 24 \, \mu\text{m} \]

### Final Answer
The thickness of each glass plate is \( 24 \, \mu\text{m} \).

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