A capillary tube of the radius r is immersed in water and water rise in it to a height H. Mass of water in the capillary tube is m. If the capillary of radius 2r is taken and dipped in water, the mass of water that will rise in the capillary tube will be

Options

A. m
B. 2 m
C. `m/2`
D. 4 m

Answer

To solve the problem, we need to understand the relationship between the radius of the capillary tube, the height of the liquid column, and the mass of the liquid that rises in the tube.

### Step-by-Step Solution:

1. **Understanding Capillary Action**:
The height \( H \) to which a liquid rises in a capillary tube is given by the formula:
\[ H = \frac{2T \cos \theta}{\rho g r} \]
where \( T \) 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, and \( r \) is the radius of the capillary tube.

2. **Mass of Water in the First Capillary Tube**:
The mass of water \( m \) in the first capillary tube of radius \( r \) can be expressed as:
\[ m = \rho V = \rho \cdot A \cdot H = \rho \cdot \pi r^2 H \]
Substituting \( H \) from the first equation:
\[ m = \rho \cdot \pi r^2 \cdot \frac{2T \cos \theta}{\rho g r} = \frac{2\pi T \cos \theta r}{g} \]

3. **Considering the Second Capillary Tube**:
For the second capillary tube with radius \( 2r \), the height \( H' \) to which the liquid will rise is given by:
\[ H' = \frac{2T \cos \theta}{\rho g (2r)} = \frac{H}{2} \]
This shows that the height of the liquid column in the second tube is half that of the first tube.

4. **Mass of Water in the Second Capillary Tube**:
The mass \( m' \) of water that will rise in the second capillary tube can be calculated as:
\[ m' = \rho \cdot \pi (2r)^2 H' = \rho \cdot \pi (2r)^2 \cdot \frac{H}{2} \]
Simplifying this gives:
\[ m' = \rho \cdot \pi \cdot 4r^2 \cdot \frac{H}{2} = 2 \rho \cdot \pi r^2 H \]

5. **Relating \( m' \) to \( m \)**:
From the earlier step, we have:
\[ m = \rho \cdot \pi r^2 H \]
Therefore, we can express \( m' \) in terms of \( m \):
\[ m' = 2m \]

### Conclusion:
The mass of water that will rise in the capillary tube of radius \( 2r \) is \( 2m \).