A microscope is focused on a mark on a piece of paper and then a slab of glass of thickness `3cm` and refractive index `1.5` is placed over the mark. How should the microscope be moved to get the mark in focus again ?

Options

A. a.` 1`cm upward
B. b. `4.5` cm downward
C. `c. 1 `cm downward
D. `d. 2` cm upward

Answer

To solve the problem step by step, we need to determine how the microscope should be adjusted after placing a glass slab over the mark on the paper.

### Step 1: Understand the problem
We have a mark on a piece of paper that is initially in focus under a microscope. When a glass slab of thickness \( t = 3 \, \text{cm} \) and refractive index \( n = 1.5 \) is placed over the mark, we need to find out how to adjust the microscope to bring the mark back into focus.

### Step 2: Use the formula for apparent depth
The relationship between the real depth (thickness of the glass slab) and the apparent depth can be given by the formula:
\[ n = \frac{\text{Real Depth}}{\text{Apparent Depth}} \]
Here, the real depth \( x = 3 \, \text{cm} \) and the refractive index \( n = 1.5 \).

### Step 3: Rearranging the formula
From the formula, we can rearrange it to find the apparent depth \( y \):
\[ y = \frac{x}{n} \]

### Step 4: Substitute the values
Substituting the known values into the equation:
\[ y = \frac{3 \, \text{cm}}{1.5} = 2 \, \text{cm} \]

### Step 5: Calculate the distance the mark appears to be raised
The distance through which the mark appears to be raised can be calculated by:
\[ \text{Distance raised} = \text{Real Depth} - \text{Apparent Depth} = x - y \]
Substituting the values:
\[ \text{Distance raised} = 3 \, \text{cm} - 2 \, \text{cm} = 1 \, \text{cm} \]

### Step 6: Determine the direction of adjustment
Since the mark appears to be raised by \( 1 \, \text{cm} \), the microscope needs to be moved upward by \( 1 \, \text{cm} \) to bring the mark back into focus.

### Conclusion
Thus, the microscope should be moved **1 centimeter upward** to get the mark in focus again.