A parallel plate capacitor is located horizontally that one of its plates is submerged in a liquid while other is above the liquid surface.The permittivity of the liquid is equal to $(epsilon).Its density is equal to >(row) .To what height will the level of liquid in the capacitor rise after its plates get a charge of surface density ~(sigma)

To determine how high the liquid level will rise in a parallel plate capacitor when one plate is submerged and the other is above the liquid, we need to consider the balance of forces acting on the liquid due to the electric field created by the charged plates. Let's break this down step by step.

Understanding the Setup

In this scenario, we have a parallel plate capacitor with one plate submerged in a liquid and the other plate exposed to air. The liquid has a permittivity denoted by ε and a density represented by ρ. When the plates are charged with a surface charge density σ, an electric field is established between the plates, which influences the liquid.

Electric Field and Pressure Relationship

The electric field (E) between the plates of a capacitor is given by the formula:

• E = σ / ε

This electric field exerts a force on the liquid, which can be understood in terms of pressure. The pressure difference created by the electric field can be expressed as:

• ΔP = ρgh

Here, ΔP is the pressure difference, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height of the liquid column that we want to find.

Balancing Forces

When the capacitor plates are charged, the electric field exerts an upward force on the liquid. This upward force must balance the weight of the liquid column above the submerged plate. Thus, we can set the pressure due to the electric field equal to the hydrostatic pressure of the liquid:

• ρgh = σE

Substituting the expression for the electric field (E) into this equation gives:

• ρgh = σ(σ / ε)

Rearranging this equation allows us to solve for the height (h) of the liquid:

• h = (σ² / (ρgε))

Final Expression for Liquid Height

Thus, the height to which the liquid will rise in the capacitor after the plates acquire a charge of surface density σ is given by:

• h = (σ² / (ρgε))

This formula shows that the height of the liquid column is directly proportional to the square of the surface charge density and inversely proportional to the product of the liquid's density, gravitational acceleration, and its permittivity.

Example Calculation

For a practical understanding, let’s consider an example. Suppose we have a liquid with a density of 1000 kg/m³ (like water), a permittivity of 80 x 8.85 x 10⁻¹² F/m (for water), and a surface charge density of 0.01 C/m². Plugging these values into our formula:

• h = (0.01² / (1000 * 9.81 * (80 * 8.85 x 10⁻¹²)))

Calculating this will give you the height to which the liquid will rise in the capacitor. This example illustrates how the physical properties of the liquid and the charge on the plates interact to determine the liquid level.

In summary, the rise of the liquid in the capacitor is a fascinating interplay of electrostatics and fluid mechanics, showcasing how electric fields can influence physical systems in practical applications.