What is the height to which a liquid rises between two long parallel plates, at d distance a part between them T and p are surface tension and density of liquid respectively and contact angle between plate and liquid is zero.

Options

A. `(4T)/(pgd)`
B. `(2T)/(pgd)`
C. `(T)/(pgd)`
D. `(T)/(2pgd)`

Explanation

To find the height to which a liquid rises between two long parallel plates, we can use the principles of capillarity. The height \( h \) of the liquid column can be derived from the balance of forces acting on the liquid.

### Step-by-Step Solution:

1. **Understand the Forces Involved**:
- The liquid will rise due to the surface tension \( T \) acting on the liquid's surface.
- The force due to surface tension can be expressed as \( F = 2T \cdot L \), where \( L \) is the length of the liquid column in contact with the plates.

2. **Balance the Forces**:
- The upward force due to surface tension must balance the downward gravitational force acting on the liquid column.
- The gravitational force can be expressed as \( F_g = m \cdot g = \rho \cdot V \cdot g \), where \( V \) is the volume of the liquid column, \( \rho \) is the density of the liquid, and \( g \) is the acceleration due to gravity.

3. **Express the Volume**:
- The volume \( V \) of the liquid column can be expressed as \( V = A \cdot h \), where \( A \) is the cross-sectional area of the liquid column and \( h \) is the height of the liquid column.
- For a narrow gap between the plates, \( A \) can be approximated as \( L \cdot d \), where \( d \) is the distance between the plates.

4. **Set Up the Equation**:
- The upward force due to surface tension is equal to the downward gravitational force:
\[ 2T \cdot L = \rho \cdot (L \cdot d \cdot h) \cdot g \]

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

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

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