The rate of flow of a liquid through a capillary tube is

Options

A. directly proportional to the length of tube.
B. inversely proportional to the difference of pressure between the ends of the tube.
C. directly proportional to the `4^(th)` power of the radius of the tube.
D. independent of the nature of the liquid.

Answer

To solve the question regarding the rate of flow of a liquid through a capillary tube, we can analyze the relationship between various factors involved in the flow.

### Step-by-Step Solution:

1. **Understanding the Formula**:
The rate of flow of a liquid (Q) through a capillary tube can be expressed by the formula:
\[ Q = \frac{\pi \Delta P R^4}{8 \eta L} \]
where:
- \( Q \) = rate of flow
- \( \Delta P \) = pressure difference between the two ends of the tube
- \( R \) = radius of the tube
- \( \eta \) = coefficient of viscosity of the liquid
- \( L \) = length of the tube

2. **Analyzing Each Option**:
- **Option A**: "Directly proportional to the length of the tube."
- From the formula, we see that \( L \) is in the denominator, which means that as the length increases, the flow rate \( Q \) decreases. Thus, this option is incorrect.

- **Option B**: "Inversely proportional to the difference of pressure between the ends of the tube."
- The formula shows that \( Q \) is directly proportional to \( \Delta P \) (as it is in the numerator). Therefore, this option is also incorrect.

- **Option C**: "Directly proportional to the fourth power of radius of the tube."
- The formula indicates that \( Q \) is directly proportional to \( R^4 \). This means that if the radius increases, the flow rate increases significantly. This option is correct.

- **Option D**: "Independent of the nature of the liquid."
- The coefficient of viscosity \( \eta \) in the formula indicates that the flow rate depends on the nature of the liquid. Therefore, this option is incorrect.

3. **Conclusion**:
Based on the analysis of the options, the correct answer is **C**: the rate of flow of a liquid through a capillary tube is directly proportional to the fourth power of the radius of the tube.