What is the height to which a liquid rises between two long parallel plates, a distance d apart ? (Surface tension of liquid is T and density is `rho`)

Options

A. `(4T)/(rho g d)`
B. `(2T)/(rho g d)`
C. `(T)/(rho g d)`
D. `(T)/(2 rho g d)`

Answer

To solve the problem of finding the height to which a liquid rises between two long parallel plates a distance \( d \) apart, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have two long parallel plates separated by a distance \( d \).
- The liquid rises to a height \( h \) between these plates due to surface tension.

2. **Identify the Forces**:
- The upward force due to surface tension acts along the perimeter of the liquid surface in contact with the plates.
- The downward force is due to the weight of the liquid column that has risen.

3. **Calculate the Upward Force**:
- The force due to surface tension \( F_T \) can be expressed as:
\[ F_T = 2T \cdot L \]
where \( T \) is the surface tension and \( L \) is the length of the plates.

4. **Calculate the Downward Force**:
- The weight of the liquid column can be expressed as:
\[ F_W = \text{Volume} \cdot \text{Density} \cdot g = h \cdot d \cdot L \cdot \rho \cdot g \]
where \( \rho \) is the density of the liquid and \( g \) is the acceleration due to gravity.

5. **Set Up the Equation**:
- At equilibrium, the upward force due to surface tension equals the downward force due to the weight of the liquid:
\[ 2T \cdot L = h \cdot d \cdot L \cdot \rho \cdot g \]

6. **Simplify the Equation**:
- We can cancel \( L \) from both sides (assuming \( L \neq 0 \)):
\[ 2T = h \cdot d \cdot \rho \cdot g \]

7. **Solve for Height \( h \)**:
- Rearranging the equation gives:
\[ h = \frac{2T}{\rho g d} \]

### Final Answer:
The height to which the liquid rises between the two plates is given by:
\[ h = \frac{2T}{\rho g d} \]