A capillary tube of radius r is dipped in a liquid of density `rho ` and surface tension S. if the angle of contanct is `theta`, the pressure difference between the two surface in the beaker and the capillary ?

Options

A. `S/r co stheta`
B. `(2S)/r cos theta`
C. `S/(r cos theta)`
D. `(2S)/(r cos theta)`

Answer

To find the pressure difference between the liquid surface in the beaker and the liquid surface in the capillary tube, we can use the concept of capillarity and the relationship between surface tension, radius, and pressure difference.

### Step-by-Step Solution:

1. **Understand the Concept of Capillarity**:
- When a capillary tube is dipped into a liquid, the liquid either rises or falls in the tube depending on the angle of contact (wetting or non-wetting).
- The height to which the liquid rises or falls in the tube is determined by the balance between the gravitational force and the force due to surface tension.

2. **Identify the Relevant Formula**:
- The pressure difference (\( \Delta P \)) between the liquid surface in the beaker and the liquid surface in the capillary tube can be expressed using the formula:
\[ \Delta P = \frac{2S \cos \theta}{r} \]
where:
- \( S \) = surface tension of the liquid,
- \( \theta \) = angle of contact,
- \( r \) = radius of the capillary tube.

3. **Relate Pressure Difference to Height**:
- The pressure difference can also be related to the height (\( h \)) of the liquid column in the capillary tube using hydrostatic pressure:
\[ \Delta P = \rho g h \]
where:
- \( \rho \) = density of the liquid,
- \( g \) = acceleration due to gravity.

4. **Equate the Two Expressions for Pressure Difference**:
- Setting the two expressions for pressure difference equal to each other gives:
\[ \frac{2S \cos \theta}{r} = \rho g h \]

5. **Solve for Height \( h \)**:
- Rearranging the equation to solve for \( h \):
\[ h = \frac{2S \cos \theta}{\rho g r} \]

6. **Final Expression for Pressure Difference**:
- The final expression for the pressure difference between the two surfaces in the beaker and the capillary tube is:
\[ \Delta P = \rho g h = \frac{2S \cos \theta}{r} \]

### Summary:
The pressure difference between the liquid surface in the beaker and the liquid surface in the capillary tube can be expressed as:
\[ \Delta P = \frac{2S \cos \theta}{r} \]