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

Explanation

To solve the problem of how to adjust the microscope after placing a glass slab over the mark, we can follow these steps:

### Step 1: Understand the Concept of Apparent Shift
When a glass slab is placed over an object, the light rays from the object will refract as they pass through the slab. This causes the object to appear to be at a different position than it actually is. The apparent shift can be calculated using the formula:

\[ \text{Shift} = T \left(1 - \frac{1}{\mu}\right) \]

where:
- \( T \) is the thickness of the glass slab,
- \( \mu \) is the refractive index of the glass.

### Step 2: Identify the Given Values
From the problem, we have:
- Thickness of the glass slab, \( T = 3 \, \text{cm} \)
- Refractive index of the glass, \( \mu = 1.5 \)

### Step 3: Substitute the Values into the Formula
Now, we can substitute the values into the formula to find the shift:

\[ \text{Shift} = 3 \left(1 - \frac{1}{1.5}\right) \]

### Step 4: Calculate the Value Inside the Parenthesis
First, calculate \( \frac{1}{1.5} \):

\[ \frac{1}{1.5} = \frac{2}{3} \approx 0.6667 \]

Now, calculate \( 1 - \frac{1}{1.5} \):

\[ 1 - 0.6667 = 0.3333 \]

### Step 5: Calculate the Shift
Now, multiply this result by the thickness \( T \):

\[ \text{Shift} = 3 \times 0.3333 = 1 \, \text{cm} \]

### Step 6: Determine the Direction of Movement
Since the mark appears to be higher due to the refraction, the microscope must be moved upward by the calculated shift to bring the mark back into focus.

### Final Answer
The microscope should be moved **1 cm upward** to get the mark in focus again.

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