The focal length of objective and eye lens of a microscope are `4 cm` and `8 cm` respectively. If the least distance of distinct vision is `24 cm` and object distance is `4.5 cm` from the objective lens, then the magnifying power of the microscope will be

Options

A. 18
B. 32
C. 64
D. 20

Explanation

To find the magnifying power of the microscope, we can follow these steps:

### Step 1: Identify the given values
- Focal length of the objective lens, \( f_o = 4 \, \text{cm} \)
- Focal length of the eye lens (eyepiece), \( f_e = 8 \, \text{cm} \)
- Least distance of distinct vision, \( D = 24 \, \text{cm} \)
- Object distance from the objective lens, \( u_o = -4.5 \, \text{cm} \) (negative because object distance is measured against the direction of the incoming light)

### Step 2: Use the lens formula for the objective lens
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
For the objective lens, we can rearrange this to find the image distance \( v_o \):
\[ \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \]
Substituting the known values:
\[ \frac{1}{4} = \frac{1}{v_o} - \frac{1}{-4.5} \]
This simplifies to:
\[ \frac{1}{4} = \frac{1}{v_o} + \frac{1}{4.5} \]

### Step 3: Solve for \( v_o \)
To solve for \( v_o \), first find a common denominator for the right side:
\[ \frac{1}{v_o} = \frac{1}{4} - \frac{1}{4.5} \]
Calculating the right side:
\[ \frac{1}{4} = \frac{4.5 - 4}{18} = \frac{0.5}{18} = \frac{1}{36} \]
Thus:
\[ \frac{1}{v_o} = \frac{1}{36} \]
So, \( v_o = 36 \, \text{cm} \).

### Step 4: Calculate the magnifying power \( M \)
The magnifying power \( M \) of the microscope is given by:
\[ M = \frac{v_o}{u_o} \times \left(1 + \frac{D}{f_e}\right) \]
Substituting the known values:
\[ M = \frac{36}{-4.5} \times \left(1 + \frac{24}{8}\right) \]
Calculating \( \frac{36}{-4.5} \):
\[ \frac{36}{-4.5} = -8 \]
Now calculate \( 1 + \frac{24}{8} = 1 + 3 = 4 \):
\[ M = -8 \times 4 = -32 \]
The magnifying power is taken as a positive value:
\[ M = 32 \]

### Final Answer
The magnifying power of the microscope is \( 32 \).

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