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 how high a liquid will rise in a coaxial cylinder made of glass immersed in a liquid with surface tension \( S \), we can follow these steps:

### Step 1: Understand the Forces Acting on the Liquid
The liquid inside the coaxial cylinder experiences two main forces:
- The upward force due to surface tension acting on the inner and outer surfaces of the cylinder.
- The downward force due to the weight of the liquid column.

### Step 2: Calculate the Upward Force Due to Surface Tension
The upward force \( F \) due to surface tension can be calculated using the formula:
\[ F = T \cdot L \]
Where \( T \) is the surface tension \( S \) and \( L \) is the total length of the contact line. This includes both the inner and outer surfaces of the cylinder:
\[ L = 2\pi R_1 + 2\pi R_2 \]
Thus, the total upward force becomes:
\[ F = S \cdot (2\pi R_1 + 2\pi R_2) \]

### Step 3: Calculate the Weight of the Liquid Column
The weight \( W \) of the liquid column can be expressed as:
\[ W = m \cdot g \]
Where \( m \) is the mass of the liquid and \( g \) is the acceleration due to gravity. The mass \( m \) can be calculated using the density \( \rho \) of the liquid and the volume \( V \) of the liquid column:
\[ m = \rho \cdot V \]
The volume \( V \) of the liquid column is the difference between the outer cylinder volume and the inner cylinder volume:
\[ V = \pi (R_2^2 - R_1^2) \cdot h \]
Thus, the weight becomes:
\[ W = \rho \cdot \pi (R_2^2 - R_1^2) \cdot h \cdot g \]

### Step 4: Set Up the Equation for Equilibrium
At equilibrium, the upward force due to surface tension equals the downward weight of the liquid:
\[ S \cdot (2\pi R_1 + 2\pi R_2) = \rho \cdot \pi (R_2^2 - R_1^2) \cdot h \cdot g \]

### Step 5: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ S \cdot (2(R_1 + R_2)) = \rho \cdot (R_2^2 - R_1^2) \cdot h \cdot g \]

### Step 6: Solve for Height \( h \)
Rearranging the equation to solve for \( h \):
\[ h = \frac{2S(R_1 + R_2)}{\rho (R_2^2 - R_1^2) g} \]

### Final Formula
Thus, the height \( h \) to which the liquid will rise in the coaxial cylinder is given by:
\[ h = \frac{2S(R_1 + R_2)}{\rho (R_2^2 - R_1^2) g} \]