Choose the correct option:
A capillary tube of radius r is lowered into a liquid of surface tension T and density `rho`. Given angle of contact `=0^(@)`.
The work done by surface tension will be

Options

A. (a) `(pi T^(2))/(rho g)`
B. (b) `(4 pi T^(2))/(rho g)`
C. (c) `(T^(2))/(rho g)`
D. (d) `(2T^(2))/(rho g)`

Answer

To solve the problem step by step, we will derive the work done by surface tension when a capillary tube is immersed in a liquid.

### Step 1: Understand the situation
We have a capillary tube of radius \( r \) immersed in a liquid with surface tension \( T \) and density \( \rho \). The angle of contact is given as \( 0^\circ \).

### Step 2: Determine the height of liquid rise in the capillary tube
The height \( H \) to which the liquid rises in the capillary tube can be calculated using the formula:
\[ H = \frac{2T \cos \theta}{r \rho g} \]
Given that the angle of contact \( \theta = 0^\circ \), we know that:
\[ \cos(0^\circ) = 1 \]
Thus, the formula simplifies to:
\[ H = \frac{2T}{r \rho g} \]

### Step 3: Calculate the work done by surface tension
The work done \( W \) by the surface tension when the liquid rises can be expressed as:
\[ W = \text{Force} \times \text{Distance} \]
The force due to surface tension acting around the circumference of the tube is given by:
\[ \text{Force} = T \times \text{Circumference} = T \times (2\pi r) \]
The distance that the liquid rises is \( H \). Therefore, the work done can be calculated as:
\[ W = T \times (2\pi r) \times H \]
Substituting the expression for \( H \):
\[ W = T \times (2\pi r) \times \left(\frac{2T}{r \rho g}\right) \]

### Step 4: Simplify the expression
Now, simplifying the expression:
\[ W = T \times (2\pi r) \times \left(\frac{2T}{r \rho g}\right) = \frac{4\pi T^2}{\rho g} \]

### Step 5: Identify the correct option
From the derived expression, we see that:
\[ W = \frac{4\pi T^2}{\rho g} \]
Thus, the correct option is:
**b. \( \frac{4\pi T^2}{\rho g} \)**.

### Summary of Steps:
1. Identify the parameters (radius \( r \), surface tension \( T \), density \( \rho \), angle of contact).
2. Use the formula for height of liquid rise in capillary action.
3. Calculate the work done using the force due to surface tension and the height.
4. Simplify the expression to find the final work done.
5. Match the result with the provided options.