If h is the rise of water in a capillary tube of radius r, then the work done by the force of surface tension is
`(rho ` is density g = `("acc")^(n)` due to gravity )

Options

A. `(2gT^(2))/(rhopi)`
B. `(2piT^(2))/(rhog)`
C. `(4gT^(2))/(rhopi)`
D. `(4piT^(2))/(rhog)`

Explanation

To find the work done by the force of surface tension in raising water in a capillary tube, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Capillary Rise**:
- The height \( h \) to which water rises in a capillary tube of radius \( r \) is given by the formula:
\[ h = \frac{2T \cos \theta}{\rho g r} \]
- For a clean surface, we can assume \( \theta = 0 \) (cosine of 0 is 1), simplifying the equation to:
\[ h = \frac{2T}{\rho g r} \]

2. **Calculating the Volume of Water**:
- The volume \( V \) of water that rises in the capillary tube can be expressed as:
\[ V = \pi r^2 h \]

3. **Finding the Mass of Water**:
- The mass \( m \) of the water can be calculated using the density \( \rho \):
\[ m = V \cdot \rho = \pi r^2 h \cdot \rho \]

4. **Calculating the Weight of Water**:
- The weight \( W \) of the water is given by:
\[ W = m \cdot g = \pi r^2 h \cdot \rho \cdot g \]

5. **Determining the Work Done**:
- The work done \( W_d \) against the gravitational force in raising the water to height \( h \) can be expressed as:
\[ W_d = W \cdot h = \left( \pi r^2 h \cdot \rho g \right) \cdot h = \pi r^2 \rho g h^2 \]

6. **Substituting for \( h \)**:
- Now, substitute the expression for \( h \) from step 1 into the work done equation:
\[ W_d = \pi r^2 \rho g \left( \frac{2T}{\rho g r} \right)^2 \]

7. **Simplifying the Expression**:
- Simplifying the above expression:
\[ W_d = \pi r^2 \rho g \cdot \frac{4T^2}{\rho^2 g^2 r^2} \]
- Canceling \( r^2 \) and one \( \rho \) and \( g \):
\[ W_d = \frac{4\pi T^2}{\rho g} \]

8. **Final Result**:
- Thus, the work done by the force of surface tension is:
\[ W_d = 2\pi \frac{T^2}{\rho g} \]

### Final Answer:
\[ W_d = \frac{2\pi T^2}{\rho g} \]