A coaxial cylinder made of glass is immersed in a-liquid of surface tension S. The radius of the inner and outer surface of the cylinder are `R_1 and R_2` respectively . Height till which liquid will rise is (Density of liquid is `rho` )

Options

A. `(2S)/(R_2rhog)`
B. `(2S)/(R_1rhog)`
C. `S/((R_2-R_1)rhog)`
D. `(2S)/((R_2-R_1)rhog)`

Answer

To solve the problem of determining the height to which a liquid will rise in a coaxial cylinder due to surface tension, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a coaxial cylinder made of glass, with an inner radius \( R_1 \) and an outer radius \( R_2 \).
- The cylinder is immersed in a liquid with surface tension \( S \) and density \( \rho \).

2. **Concept of Capillarity**:
- The phenomenon of liquid rising in a narrow space due to surface tension is known as capillarity.
- The height \( h \) to which the liquid rises in a capillary tube is given by the formula:
\[ h = \frac{2S \cos \theta}{\rho g r} \]
where:
- \( S \) = surface tension of the liquid,
- \( \theta \) = angle of contact,
- \( \rho \) = density of the liquid,
- \( g \) = acceleration due to gravity,
- \( r \) = effective radius of the capillary.

3. **Determining the Effective Radius**:
- In this case, the effective radius \( r \) is the difference between the outer and inner radii of the cylinder:
\[ r = R_2 - R_1 \]

4. **Angle of Contact**:
- For a glass surface in contact with water, the angle of contact \( \theta \) is \( 0^\circ \).
- Therefore, \( \cos \theta = \cos 0^\circ = 1 \).

5. **Substituting Values into the Formula**:
- Substitute \( r \) and \( \cos \theta \) into the height formula:
\[ h = \frac{2S \cdot 1}{\rho g (R_2 - R_1)} \]
- This simplifies to:
\[ h = \frac{2S}{\rho g (R_2 - R_1)} \]

6. **Final Result**:
- The height \( h \) to which the liquid will rise in the coaxial cylinder is given by:
\[ h = \frac{2S}{\rho g (R_2 - R_1)} \]