Two vertical parallel glass plates are partially submerged in water. The distance between the plates is d and the length is `l`. Assume that the water between the plates does not reach the upper edges of the plates and the plates and the wetting is complete. The water will rise to height (`rho=` density of water and `alpha =` surface tension of water)

Options

A. `(2sigma)/(rhogd)`
B. `(sigma)/(2rhogd)`
C. `(4sigma)/(rhogd)`
D. `(5sigma)/(rhogd)`

Explanation

To solve the problem of determining the height to which water will rise between two vertical parallel glass plates partially submerged in water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Water:**
- The water between the plates experiences two main forces: the upward force due to surface tension and the downward force due to the weight of the water column.

2. **Calculate the Surface Tension Force:**
- The surface tension force (F_s) acting on the water column between the plates can be expressed as:
\[ F_s = 2 \sigma L \]
where \( \sigma \) is the surface tension of water and \( L \) is the length of the plates.

3. **Calculate the Weight of the Water Column:**
- The weight (W) of the water column can be calculated using the formula:
\[ W = \rho V g \]
where \( \rho \) is the density of water, \( V \) is the volume of water between the plates, and \( g \) is the acceleration due to gravity. The volume \( V \) can be expressed as:
\[ V = l \cdot d \cdot h \]
where \( d \) is the distance between the plates and \( h \) is the height of the water column.

4. **Set Up the Equation:**
- At equilibrium, the upward surface tension force equals the downward weight of the water column:
\[ 2 \sigma L = \rho (l \cdot d \cdot h) g \]

5. **Simplify the Equation:**
- We can cancel \( L \) from both sides of the equation:
\[ 2 \sigma = \rho d h g \]

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

7. **Final Result:**
- The height to which water will rise between the plates is given by:
\[ h = \frac{2 \sigma}{\rho d g} \]