A glass rod of radius `r_(1)` is inserted symmetrically into a vertical capillary tube of radius `r_(2)` such that their lower ends are at the same level. The arrangement is now dipped in water. The height to which water will rise into the tube will be (`sigma =` surface tension of water, `rho = ` density of water)

Options

A. `(2sigma)/((r_(2)-r_(1))rhog)`
B. `sigma/((r_(2)-r_(1))rhog)`
C. `(2sigma)/((r_(2)+r_(1))rhog)`
D. `(2sigma)/((r_(2)^(2)+r_(1)^(2))rhog)`

Explanation

To solve the problem of how high water will rise in a capillary tube when a glass rod is inserted into it, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces Involved**:
- When the glass rod is inserted into the capillary tube, the surface tension of water acts at the interface between the water and the glass rod. This creates an upward force that will cause the water to rise in the tube.

2. **Calculating the Upward Force due to Surface Tension**:
- The total upward force \( F \) due to surface tension can be calculated as:
\[ F = 2 \sigma \pi r_1 + 2 \sigma \pi r_2 \]
- Here, \( \sigma \) is the surface tension of water, \( r_1 \) is the radius of the glass rod, and \( r_2 \) is the radius of the capillary tube.

3. **Calculating the Weight of the Water Column**:
- The weight \( W \) of the water column of height \( h \) that rises in the tube can be expressed as:
\[ W = h \cdot \pi (r_2^2 - r_1^2) \cdot \rho g \]
- In this equation, \( \rho \) is the density of water, and \( g \) is the acceleration due to gravity.

4. **Setting Up the Equation**:
- At equilibrium, the upward force due to surface tension will equal the weight of the water column:
\[ 2 \sigma \pi r_1 + 2 \sigma \pi r_2 = h \cdot \pi (r_2^2 - r_1^2) \cdot \rho g \]

5. **Simplifying the Equation**:
- We can cancel \( \pi \) from both sides:
\[ 2 \sigma (r_1 + r_2) = h (r_2^2 - r_1^2) \rho g \]

6. **Solving for Height \( h \)**:
- Rearranging the equation to solve for \( h \):
\[ h = \frac{2 \sigma (r_1 + r_2)}{(r_2^2 - r_1^2) \rho g} \]

### Final Answer:
The height to which water will rise in the tube is given by:
\[ h = \frac{2 \sigma (r_1 + r_2)}{(r_2^2 - r_1^2) \rho g} \]