Water rises to a height of 30 mm in a capillary tube. If the radius of the capillary tube is made `3//4` of its previous value. The height to which the water will rise in the tube is

Options

A. 30 mm
B. 20 mm
C. 40 mm
D. 10 mm

Explanation

To solve the problem of how high water will rise in a capillary tube when the radius is changed, we can use the formula for capillary rise:

\[ h = \frac{2T \cos \theta}{R \rho g} \]

Where:
- \( h \) is the height of the liquid column,
- \( T \) is the surface tension of the liquid,
- \( \theta \) is the contact angle,
- \( R \) is the radius of the capillary tube,
- \( \rho \) is the density of the liquid,
- \( g \) is the acceleration due to gravity.

### Step-by-step Solution:

1. **Identify the initial conditions**:
- Given that the initial height of water rise, \( h_1 = 30 \) mm.
- Let the initial radius of the capillary tube be \( r_1 = R \).

2. **Determine the new radius**:
- The new radius \( r_2 \) is \( \frac{3}{4} \) of the initial radius, so:
\[ r_2 = \frac{3}{4} R \]

3. **Use the relationship between height and radius**:
- From the capillary rise formula, we know that the product of height and radius is constant for a given liquid:
\[ h_1 r_1 = h_2 r_2 \]

4. **Substitute the known values**:
- Substitute \( h_1 = 30 \) mm, \( r_1 = R \), and \( r_2 = \frac{3}{4} R \) into the equation:
\[ 30 \cdot R = h_2 \cdot \left(\frac{3}{4} R\right) \]

5. **Solve for \( h_2 \)**:
- Cancel \( R \) from both sides (assuming \( R \neq 0 \)):
\[ 30 = h_2 \cdot \frac{3}{4} \]
- Rearranging gives:
\[ h_2 = 30 \cdot \frac{4}{3} \]
- Calculate \( h_2 \):
\[ h_2 = 40 \text{ mm} \]

### Final Answer:
The height to which the water will rise in the tube is **40 mm**.