Why a liquid rises more in a capillary with narrow bore than same with wider bore?

Answer

To understand why a liquid rises more in a capillary tube with a narrow bore compared to one with a wider bore, we can analyze the situation using the formula for capillary action.

### Step-by-Step Solution:

1. **Understanding Capillary Action**:
- Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces. It occurs due to the combination of cohesive forces (attraction between like molecules) and adhesive forces (attraction between unlike molecules).

2. **Formula for Capillary Rise**:
- The height (h) to which a liquid rises in a capillary tube can be described by the formula:
\[ h = \frac{2T \cos \theta}{\rho r g} \]
Where:
- \( T \) = surface tension of the liquid
- \( \theta \) = contact angle
- \( \rho \) = density of the liquid
- \( r \) = radius of the capillary tube
- \( g \) = acceleration due to gravity

3. **Analyzing the Formula**:
- From the formula, we can see that the height \( h \) is inversely proportional to the radius \( r \) of the capillary tube. This means that as the radius increases, the height \( h \) decreases, and vice versa.

4. **Comparing Narrow and Wide Tubes**:
- In a narrow capillary tube, the radius \( r \) is small, which leads to a larger height \( h \) of the liquid column.
- In a wider capillary tube, the radius \( r \) is larger, which results in a smaller height \( h \) of the liquid column.

5. **Conclusion**:
- Therefore, a liquid rises more in a capillary tube with a narrow bore compared to a wider bore because the height of the liquid column is inversely proportional to the radius of the tube. Thus, the narrower the tube, the higher the liquid rises.