The focal length of the objective of a compound microscope is `f_0` and its distance from the eyepiece is L. The object is placed at a distance u from the objective. For proper working of the instrument.,

Options

A. `Lltu`
B. `Lgtu`
C. `f_0ltLlt2f_0`
D. `Lgt2f_0`

Answer

To solve the problem regarding the proper working of a compound microscope, we need to analyze the relationships between the focal length of the objective lens (f₀), the distance between the objective and the eyepiece (L), and the object distance (u).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A compound microscope consists of two lenses: the objective lens and the eyepiece. The objective lens forms a real image of the object placed at distance u.
- The distance between the objective lens and the eyepiece is denoted as L.

2. **Image Formation by the Objective Lens**:
- The objective lens forms a real image (I₁) of the object (O). For the objective lens, the lens formula is given by:
\[ \frac{1}{f_0} = \frac{1}{v_1} - \frac{1}{u} \]
where v₁ is the image distance from the objective lens.

3. **Position of the Image**:
- For proper functioning of the compound microscope, the image I₁ must be located on the left side of the eyepiece (i.e., before the eyepiece). This means that the image formed by the objective lens must be within the focal length of the eyepiece.

4. **Condition for Image Formation**:
- The image I₁ formed by the objective lens must be at a distance v₁ such that:
\[ v₁ < L \]
- This implies that the distance L must be greater than the distance u (the object distance) because the real image formed by the objective lens must be closer to the eyepiece than the distance between the two lenses.

5. **Analyzing the Options**:
- From our analysis, we can derive that:
- **Condition 1**: \( L > u \)
- **Condition 2**: The distance L must also be greater than twice the focal length of the objective lens to ensure that the image is formed properly and can be viewed through the eyepiece:
\[ L > 2f_0 \]

6. **Conclusion**:
- Therefore, for the proper working of the compound microscope, the following conditions must hold true:
- \( L > u \)
- \( L > 2f_0 \)

### Summary of Correct Options:
- The correct options based on our analysis are:
- \( L > u \)
- \( L > 2f_0 \)