Two capillary tubes P and Q are dipped in water. The height of water level in capillary P is 2/3 to the height in Q capillary. The ratio of their diameters is

Options

A. `2:3`
B. `3:2`
C. `3:4`
D. `4:3`

Answer

To solve the problem, we need to analyze the relationship between the height of the water column in the capillary tubes and their diameters.

### Step-by-Step Solution:

1. **Understanding Capillary Action**:
The height of the liquid column (H) in a capillary tube is given by the formula:
\[ H = \frac{2T \cos \theta}{R \rho g} \]
where:
- \( T \) = surface tension of the liquid,
- \( \theta \) = angle of contact,
- \( R \) = radius of the capillary tube,
- \( \rho \) = density of the liquid,
- \( g \) = acceleration due to gravity.

2. **Relating Heights in Tubes P and Q**:
Given that the height of water in capillary tube P (denoted as \( H_P \)) is \( \frac{2}{3} \) of the height in capillary tube Q (denoted as \( H_Q \)), we can write:
\[ H_P = \frac{2}{3} H_Q \]

3. **Setting Up the Ratios**:
From the relationship established, we can express the ratio of the heights:
\[ \frac{H_P}{H_Q} = \frac{2}{3} \]

4. **Inversely Proportional Relationship**:
The height of the liquid column in a capillary tube is inversely proportional to the radius of the tube:
\[ H \propto \frac{1}{R} \]
This means:
\[ \frac{H_P}{H_Q} = \frac{R_Q}{R_P} \]

5. **Substituting the Ratios**:
Using the ratio of heights:
\[ \frac{2}{3} = \frac{R_Q}{R_P} \]

6. **Finding the Ratio of Radii**:
Rearranging gives us:
\[ \frac{R_P}{R_Q} = \frac{3}{2} \]

7. **Finding the Ratio of Diameters**:
Since the diameter \( D \) is twice the radius \( R \):
\[ D_P = 2R_P \quad \text{and} \quad D_Q = 2R_Q \]
Therefore, the ratio of the diameters is:
\[ \frac{D_P}{D_Q} = \frac{R_P}{R_Q} = \frac{3}{2} \]

### Final Answer:
The ratio of the diameters of capillary tubes P and Q is:
\[ D_P : D_Q = 3 : 2 \]