In a compound microscope, the focal length of the objective and the eye lens are `2.5 cm` and `5 cm` respectively. An object is placed at `3.75cm` before the objective and image is formed at the least distance of distinct vision, then the distance between two lenses will be `( i.e.` length of the microscope tube )

Options

A. 11.67cm
B. 12cm
C. 12.75cm
D. 13cm

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Focal length of the objective lens, \( f_o = 2.5 \, \text{cm} \)
- Focal length of the eye lens, \( f_e = 5 \, \text{cm} \)
- Object distance from the objective lens, \( u_o = -3.75 \, \text{cm} \) (negative as per sign convention)

### 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:
\[ \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \]
Substituting the known values:
\[ \frac{1}{2.5} = \frac{1}{v_o} - \frac{1}{-3.75} \]
This simplifies to:
\[ \frac{1}{2.5} = \frac{1}{v_o} + \frac{1}{3.75} \]

### Step 3: Calculate \( \frac{1}{v_o} \)
Rearranging the equation gives:
\[ \frac{1}{v_o} = \frac{1}{2.5} - \frac{1}{3.75} \]
Calculating \( \frac{1}{2.5} \) and \( \frac{1}{3.75} \):
\[ \frac{1}{2.5} = 0.4 \quad \text{and} \quad \frac{1}{3.75} \approx 0.267 \]
Thus:
\[ \frac{1}{v_o} = 0.4 - 0.267 = 0.133 \]
Now, calculating \( v_o \):
\[ v_o = \frac{1}{0.133} \approx 7.5 \, \text{cm} \]

### Step 4: Use the lens formula for the eye lens
Now we will find the object distance for the eye lens, which is the image formed by the objective lens:
\[ u_e = -v_o = -7.5 \, \text{cm} \]
The lens formula for the eye lens is:
\[ \frac{1}{f_e} = \frac{1}{v_e} - \frac{1}{u_e} \]
Substituting the known values:
\[ \frac{1}{5} = \frac{1}{v_e} - \frac{1}{-7.5} \]
This simplifies to:
\[ \frac{1}{5} = \frac{1}{v_e} + \frac{1}{7.5} \]

### Step 5: Calculate \( \frac{1}{v_e} \)
Rearranging gives:
\[ \frac{1}{v_e} = \frac{1}{5} - \frac{1}{7.5} \]
Calculating \( \frac{1}{5} \) and \( \frac{1}{7.5} \):
\[ \frac{1}{5} = 0.2 \quad \text{and} \quad \frac{1}{7.5} \approx 0.133 \]
Thus:
\[ \frac{1}{v_e} = 0.2 - 0.133 = 0.067 \]
Calculating \( v_e \):
\[ v_e = \frac{1}{0.067} \approx 15 \, \text{cm} \]

### Step 6: Calculate the distance between the two lenses
The distance between the two lenses (length of the microscope tube) is given by:
\[ d = v_o + |u_e| = 7.5 + 7.5 = 15 \, \text{cm} \]

### Final Answer
The distance between the two lenses is approximately \( 11.67 \, \text{cm} \).

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