The magnifying power of a compound microscope is high if

Options

A. both eye-piece and objective have short focal lengths
B. both objective and eye-piece have long focal lengths
C. the objective has a long focal length and eye-piece has a short focal length
D. the objective has a short focal length and the eye-piece has a long focal length

Answer

To determine when the magnifying power of a compound microscope is high, we need to analyze the factors that affect magnification. The magnifying power (M) of a compound microscope is given by the formula:

\[ M = \frac{v_o}{u_o} \left(1 + \frac{d}{f_e}\right) \]

Where:
- \( v_o \) = image distance from the objective lens
- \( u_o \) = object distance from the objective lens
- \( d \) = near point distance (usually taken as 25 cm for a normal human eye)
- \( f_e \) = focal length of the eyepiece lens
- \( f_o \) = focal length of the objective lens

### Step-by-step Solution:

1. **Understanding the Components**:
- The compound microscope consists of two lenses: the objective lens and the eyepiece lens. The objective lens creates a magnified image of the object, which is then viewed through the eyepiece.

2. **Focal Lengths**:
- The focal lengths of both the objective lens (\( f_o \)) and the eyepiece lens (\( f_e \)) play a crucial role in determining the magnification.
- A shorter focal length for either lens results in a greater magnification.

3. **Analyzing the Formula**:
- From the formula, we see that the magnification \( M \) increases if:
- \( u_o \) (the object distance) is minimized, which can be achieved by placing the object close to the lens.
- \( f_e \) (the focal length of the eyepiece) is minimized, which means using an eyepiece with a short focal length.
- The term \( \frac{d}{f_e} \) indicates that a smaller \( f_e \) increases the magnification.

4. **Evaluating the Options**:
- **Option A**: Both eyepiece and objective lens have short focal lengths. This is correct because shorter focal lengths increase magnification.
- **Option B**: Both objective and eyepiece have long focal lengths. This is incorrect as it would decrease magnification.
- **Option C**: The objective has a long focal length and the eyepiece has a short focal length. This is incorrect because the long focal length of the objective would reduce magnification.
- **Option D**: The objective lens has a short focal length and the eyepiece has a long focal length. This is also incorrect as the long focal length of the eyepiece would reduce overall magnification.

5. **Conclusion**:
- The correct answer is **Option A**: Both eyepiece and objective lens have short focal lengths, which maximizes the magnifying power of the compound microscope.