A capillary tube of radius R is immersed in water and water rises in it to a height H . Mass of water in the capillary tube is . M If the radius of the tube is doubled, mass of water that will rise in the capillary tube will now be

Options

A. 2 M
B. M
C. 8 M
D. 4 M

Answer

To solve the problem, we need to analyze how the mass of water in a capillary tube changes when the radius of the tube is doubled.

### Step-by-Step Solution:

1. **Understand the Capillary Rise Formula**:
The height \( H \) to which water rises in a capillary tube is given by the formula:
\[ H = \frac{2 \sigma \cos \theta}{\rho g r} \]
where:
- \( \sigma \) 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.

2. **Calculate the Volume of Water in the Tube**:
The volume \( V \) of water that rises in the capillary tube can be expressed as:
\[ V = \text{Area} \times \text{Height} = \pi r^2 H \]
Substituting the expression for \( H \):
\[ V = \pi r^2 \left(\frac{2 \sigma \cos \theta}{\rho g r}\right) = \frac{2 \pi \sigma \cos \theta}{\rho g} r \]

3. **Calculate the Mass of Water**:
The mass \( M \) of the water in the capillary tube is given by:
\[ M = \rho V = \rho \left(\frac{2 \pi \sigma \cos \theta}{\rho g} r\right) = \frac{2 \pi \sigma \cos \theta}{g} r \]

4. **Analyze the Effect of Doubling the Radius**:
If the radius of the capillary tube is doubled (i.e., \( r \) becomes \( 2r \)), we can substitute this into the mass equation:
\[ M' = \frac{2 \pi \sigma \cos \theta}{g} (2r) = \frac{4 \pi \sigma \cos \theta}{g} r \]

5. **Relate the New Mass to the Original Mass**:
From the original mass \( M = \frac{2 \pi \sigma \cos \theta}{g} r \), we can see that:
\[ M' = 2M \]
Therefore, when the radius is doubled, the mass of water that rises in the capillary tube becomes:
\[ M' = 2M \]

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