Three objective focal lengths (f) and two eye piece focal lengths (f) are available for a compound microscope. By combining these two, the magnification of microscope will be maximum when:

Options

A. `f_(0)=f_(e)`
B. `f_(0)gt gtf_(e)`
C. `f_(0)` and `f_(e)` both are small
D. `f_(0) gt f_(e)`

Answer

To solve the problem regarding the maximum magnification of a compound microscope when using different focal lengths for the objective and eyepiece, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Magnification in a Compound Microscope**:
The total magnification (M) of a compound microscope is given by the product of the magnification produced by the objective lens (Mo) and the eyepiece lens (Me):
\[ M = Mo \times Me \]

2. **Magnification of the Objective Lens**:
The magnification produced by the objective lens is given by:
\[ Mo = \frac{L}{Fo} \]
where \(L\) is the tube length and \(Fo\) is the focal length of the objective lens.

3. **Magnification of the Eyepiece Lens**:
The magnification produced by the eyepiece lens is given by:
\[ Me = 1 + \frac{D}{Fe} \]
where \(D\) is the near point distance (usually taken as 25 cm for a normal eye) and \(Fe\) is the focal length of the eyepiece lens.

4. **Analyzing the Conditions for Maximum Magnification**:
To maximize the total magnification \(M\), we need to consider the focal lengths of both the objective and eyepiece lenses.

5. **Considering the Options**:
- **Option 1**: \(Fo = Fe\) - This does not necessarily maximize magnification.
- **Option 2**: \(Fo > Fe\) - This could potentially work, but it depends on the specific values.
- **Option 3**: Both \(Fo\) and \(Fe\) are very small - This is a strong candidate since smaller focal lengths generally lead to higher magnification.
- **Option 4**: \(Fo\) is better than \(Fe\) - This is vague and does not provide a clear condition.

6. **Conclusion**:
From the analysis, we find that the maximum magnification occurs when both the objective and eyepiece focal lengths are very small. Therefore, the correct answer is:
\[ \text{Option 3: } Fo \text{ and } Fe \text{ are very small.} \]