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. `1 cm` upward
B. `4.5 cm` downward
C. `1 cm` downward
D. `2 cm` upward

Explanation

To solve the problem of how to adjust the microscope after placing a glass slab over the mark, we need to calculate the apparent depth of the mark when viewed through the glass slab. Here’s a step-by-step solution:

### Step 1: Understand the situation
The microscope is initially focused on a mark on a piece of paper. When a glass slab of thickness \( t = 3 \, \text{cm} \) and refractive index \( \mu = 1.5 \) is placed over the mark, the apparent depth of the mark changes.

### Step 2: Use the formula for apparent depth
The formula for apparent depth \( d' \) when viewing through a medium is given by:
\[ d' = \frac{d}{\mu} \]
where \( d \) is the actual depth (thickness of the slab) and \( \mu \) is the refractive index of the slab.

### Step 3: Substitute the values
Here, the actual depth \( d = 3 \, \text{cm} \) and the refractive index \( \mu = 1.5 \). Plugging in these values:
\[ d' = \frac{3 \, \text{cm}}{1.5} = 2 \, \text{cm} \]

### Step 4: Determine the change in position
Initially, the microscope was focused on the mark at an actual depth of \( 3 \, \text{cm} \). After placing the slab, the apparent depth is now \( 2 \, \text{cm} \).

### Step 5: Calculate the distance to move the microscope
The microscope must be moved to focus on the new apparent depth. The difference in depth is:
\[ \text{Distance to move} = \text{Actual depth} - \text{Apparent depth} = 3 \, \text{cm} - 2 \, \text{cm} = 1 \, \text{cm} \]

### Step 6: Conclusion
To get the mark in focus again, the microscope should be moved **1 cm closer** to the mark.

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