Two parallel glass plates are dipped partly in the liquid of density 'd' keeping them vertical. If the distance between the plates is 'x', Surface tension is T and angle of contact is `theta` then rise of liquid between the plates due to capillary will be

Options

A. `(T cos theta)/(xd)`
B. `(2T cos theta)/(xdg)`
C. `(2T)/(xdg cos theta)`
D. `(t cos theta)/(xdg)`

Explanation

To solve the problem of the rise of liquid between two parallel glass plates due to capillary action, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two parallel glass plates that are partially immersed in a liquid of density \( d \).
- The distance between the plates is \( x \).
- The surface tension of the liquid is \( T \) and the angle of contact is \( \theta \).

2. **Visualizing the Capillary Action**:
- When the plates are close enough, the liquid will rise between them due to capillary action.
- Let’s denote the height of the liquid rise as \( h \).

3. **Forces Acting on the Liquid Column**:
- The surface tension acts along the edges of the plates and creates an upward force.
- The weight of the liquid column acts downward.

4. **Calculating the Upward Force Due to Surface Tension**:
- The upward force due to surface tension can be expressed as:
\[ F_{\text{up}} = 2T \cos(\theta) \cdot L \]
- Here, \( L \) is the length of the plates in contact with the liquid.

5. **Calculating the Downward Force Due to Weight**:
- The weight of the liquid column can be expressed as:
\[ F_{\text{down}} = \text{mass} \cdot g = (\text{density} \cdot \text{volume}) \cdot g \]
- The volume of the liquid column is \( L \cdot x \cdot h \), so:
\[ F_{\text{down}} = d \cdot (L \cdot x \cdot h) \cdot g \]

6. **Setting Up the Equation**:
- For the liquid to be in equilibrium, the upward force must equal the downward force:
\[ 2T \cos(\theta) \cdot L = d \cdot (L \cdot x \cdot h) \cdot g \]

7. **Simplifying the Equation**:
- We can cancel \( L \) from both sides (assuming \( L \neq 0 \)):
\[ 2T \cos(\theta) = d \cdot x \cdot h \cdot g \]

8. **Solving for the Height \( h \)**:
- Rearranging the equation to solve for \( h \):
\[ h = \frac{2T \cos(\theta)}{d \cdot x \cdot g} \]

### Final Result:
The rise of the liquid between the plates due to capillary action is given by:
\[ h = \frac{2T \cos(\theta)}{d \cdot x \cdot g} \]