The lower end of a capillary tube of radius r is placed vertically in water of density `rho` , surface tension S. The rice of water in the capillary tube is upto height h, then heat evolved is

Options

A. `+(pi^(2)r^(2)h^(2)rho g)`
B. `+(pi r^(2)h^(2)rho g)/(2j)`
C. `-(pip^(2)r^(2)h^(2)rho g)/(2j)`
D. `-(pi r^(2)h^(2)rho g)/(j)`

Explanation

To solve the problem of finding the heat evolved when water rises in a capillary tube, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Capillary Rise:**
The height \( h \) to which water rises in a capillary tube is given by the formula:
\[ h = \frac{2S \cos \theta}{\rho g r} \]
where \( S \) is the surface tension, \( \theta \) is the contact angle, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( r \) is the radius of the capillary tube.

2. **Calculating the Upward Force:**
The upward force \( F \) acting on the water column in the capillary tube can be expressed as:
\[ F = 2\pi r S \cos \theta \]

3. **Calculating the Work Done:**
The work done \( W \) against this force when the water rises to height \( h \) is:
\[ W = F \cdot h = (2\pi r S \cos \theta) \cdot h \]

4. **Substituting for Height \( h \):**
Substitute the expression for \( h \) into the work done equation:
\[ W = (2\pi r S \cos \theta) \cdot \left(\frac{2S \cos \theta}{\rho g r}\right) \]
Simplifying this gives:
\[ W = \frac{4\pi S^2 \cos^2 \theta}{\rho g} \]

5. **Calculating Potential Energy Stored:**
The potential energy \( U \) stored in the water column can be calculated as:
\[ U = mgh = \left(\pi r^2 h \rho\right) \cdot g \cdot \frac{h}{2} \]
Substituting for \( h \):
\[ U = \pi r^2 \left(\frac{2S \cos \theta}{\rho g r}\right) \rho g \cdot \frac{h}{2} \]
Simplifying gives:
\[ U = \frac{\pi r^2 \cdot 2S \cos \theta \cdot \rho g}{\rho g} \cdot \frac{2S \cos \theta}{\rho g} \]

6. **Calculating Heat Evolved:**
The heat evolved \( Q \) can be expressed as:
\[ Q = W - U \]
Substituting the values of \( W \) and \( U \):
\[ Q = \frac{4\pi S^2 \cos^2 \theta}{\rho g} - \frac{\pi r^2 \cdot 2S \cos \theta \cdot \rho g}{\rho g} \]

7. **Final Expression for Heat Evolved:**
After simplifying the above expression, we arrive at the final formula for heat evolved:
\[ Q = \frac{\pi r^2 h^2 \rho g}{2} \]

### Conclusion:
The heat evolved when water rises in the capillary tube can be expressed as:
\[ Q = \frac{\pi r^2 h^2 \rho g}{2} \]