Water rises to a height h when a capillary of radius r is dipped in water. For another capillary of radius 2r dipped in water, it rises to the height

Options

A. a. h
B. b.2h
C. c.`(h)/(4)`
D. d. `(h)/(2)`

Answer

To solve the problem of how high water rises in a capillary tube of radius \(2r\) compared to a capillary tube of radius \(r\), we can use the formula for capillary rise, which is given by:

\[ h = \frac{2S \cos \theta}{\rho g r} \]

where:
- \(h\) is the height of the liquid column,
- \(S\) is the surface tension of the liquid,
- \(\theta\) is the contact angle,
- \(\rho\) is the density of the liquid,
- \(g\) is the acceleration due to gravity,
- \(r\) is the radius of the capillary tube.

### Step-by-Step Solution

1. **Identify the height for the first capillary tube**:
For the first capillary tube with radius \(r\), the height of the water column is given as \(h\). We can express this as:
\[ h_1 = \frac{2S \cos \theta}{\rho g r} \]
Since \(h_1 = h\), we can write:
\[ h = \frac{2S \cos \theta}{\rho g r} \]

2. **Identify the height for the second capillary tube**:
For the second capillary tube with radius \(2r\), we can express the height of the water column as:
\[ h_2 = \frac{2S \cos \theta}{\rho g (2r)} \]

3. **Simplify the expression for \(h_2\)**:
Substituting the expression for \(h_2\):
\[ h_2 = \frac{2S \cos \theta}{\rho g (2r)} = \frac{1}{2} \cdot \frac{2S \cos \theta}{\rho g r} \]
This simplifies to:
\[ h_2 = \frac{1}{2} h \]

4. **Conclusion**:
Therefore, the height to which water rises in the capillary tube of radius \(2r\) is:
\[ h_2 = \frac{h}{2} \]

### Final Answer
The height to which water rises in the second capillary tube of radius \(2r\) is \(\frac{h}{2}\).