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 determine the conditions under which the magnification of a compound microscope is maximized when using three objective focal lengths (f₀) and two eyepiece focal lengths (fₑ), we can follow these steps:

### Step 1: Understand the Magnification Formula
The total magnification (M) of a compound microscope is given by the product of the magnification produced by the objective lens (M₀) and the eyepiece (Mₑ).

- The magnification due to the objective lens is given by:
\[ M₀ = \frac{L}{f₀} \]
where L is the length of the microscope tube and f₀ is the focal length of the objective lens.

- The magnification due to the eyepiece is given by:
\[ Mₑ = \frac{D}{fₑ} \]
where D is the distance of minimum distinct vision and fₑ is the focal length of the eyepiece.

### Step 2: Combine the Magnifications
The total magnification of the microscope can be expressed as:
\[ M = M₀ \times Mₑ = \left(\frac{L}{f₀}\right) \times \left(\frac{D}{fₑ}\right) \]
This can be simplified to:
\[ M = \frac{L \cdot D}{f₀ \cdot fₑ} \]

### Step 3: Analyze the Denominators
To maximize the magnification (M), we need to minimize the denominators \(f₀\) and \(fₑ\). Since both \(f₀\) and \(fₑ\) are focal lengths, smaller values will yield a larger magnification.

### Step 4: Choose the Optimal Focal Lengths
Given that we have three objective focal lengths and two eyepiece focal lengths, we should choose the smallest available values for both \(f₀\) and \(fₑ\).

- Let’s denote the smallest objective focal length as \(f₀_{min}\) and the smallest eyepiece focal length as \(fₑ_{min}\).

### Step 5: Conclusion
The magnification of the microscope will be maximum when the smallest focal lengths for both the objective and eyepiece are used. Thus, the optimal combination is:
- Use the smallest focal length from the three available objective lenses.
- Use the smallest focal length from the two available eyepieces.