Question

A 5.80-$\mu$F, parallel-plate, air capacitor has a plate separation of 5.00 mm and is charged to a potential difference of 400 V. Calculate the energy density in the region between the plates, in units of J/m${}^3$.

Short answer

The energy density is approximately 0.0283 J/m³.

Step-by-step solution

Identify the Formula for Energy Density

The energy density (u) in the region between the plates of a capacitor is given by the formula: $$u=\frac{1}{2}{\epsilon }_0{E}^2$$ where ${\epsilon }_0$ is the permittivity of free space ($8.85\times {10}^{-12}{\text{C}}^2/(\text{N}\cdot {\text{m}}^2)$) and $E$ is the electric field strength between the plates.

Calculate the Electric Field

The electric field $E$ between parallel plates is given by the formula: $$E=\frac{V}{d}$$ where $V$ is the potential difference across the plates (400 V) and $d$ is the separation between the plates (5.00 mm or 0.005 m). Substitute these values into the formula: $$E=\frac{400\text{V}}{0.005\text{m}}=80,000\text{V/m}$$.

Substitute into the Energy Density Formula

Substitute the value of $E$ and ${\epsilon }_0$ into the energy density formula: $$u=\frac{1}{2}\times 8.85\times {10}^{-12}{\text{C}}^2/(\text{N}\cdot {\text{m}}^2)\times (80,000\text{V/m}{)}^2$$.

Compute the Energy Density

Calculate the numerical value of the energy density: $$u=\frac{1}{2}\times 8.85\times {10}^{-12}\times 6.4\times {10}^9=2.83\times {10}^{-2}{\text{J/m}}^3$$.

Key concepts

Parallel-Plate Capacitor

A parallel-plate capacitor is a simple device that consists of two flat, conductive plates separated by a distance.
It stores electrical energy by maintaining a voltage across the plates. This arrangement sets an electric field in the space between them, creating a storage field for energy.
Parallel-plate capacitors are often used in electronics because of their straightforward design and ability to store an electric charge efficiently.
When a voltage is applied, the capacitor charges up with positive charges on one plate and negative charges on the other, with an electric field established between them.

• The stored energy in a capacitor is crucial for controlling power supply and processing electronic signals in circuits.
• The capacitance, which indicates how much charge it can store, is influenced by the plate area, separation distance, and the medium between the plates.

Understanding these basics aids in grasping how capacitors function within larger electrical systems.

Electric Field Calculation

The electric field in a parallel-plate capacitor is uniform and calculated using the voltage across the plates and the distance separating them.
This relation is given by the formula: $$E=\frac{V}{d}$$
where $E$ is the electric field strength, $V$ is the potential difference, and $d$ is the plate separation.
In the context of the exercise, for a voltage of 400 V and a plate separation of 0.005 m (converted from 5.00 mm), the electric field can be calculated by substituting the values:
$$E=\frac{400\text{V}}{0.005\text{m}}=80,000\text{V/m}$$.

• Knowing the electric field helps understand the force exerted on charges between the plates.
• It's a key factor in determining the energy density—a measure of energy stored per unit volume in the capacitor.

This calculation highlights the strong influence of plate separation on the electric field and energy storage.

Permittivity of Free Space

The permittivity of free space, also known as ${\epsilon }_0$, is a fundamental physical constant that describes how electric fields interact with a vacuum.
Its value is approximately $8.85\times {10}^{-12}{\text{C}}^2/(\text{N}\cdot {\text{m}}^2)$. This constant is crucial when analyzing capacitors and their ability to store energy.
In the formula for the energy density of a parallel-plate capacitor, $u=\frac{1}{2}{\epsilon }_0{E}^2$, ${\epsilon }_0$ acts as a scaling factor, affecting how much energy is stashed between the plates for a given electric field.
The permittivity of free space defines the influence of a vacuum environment on the electric field strength and energy storage efficiency.

• Higher permittivity means greater capacitance and energy storage capability in a capacitor.
• It also plays an essential role in determining the propagation of electromagnetic waves.

Understanding ${\epsilon }_0$ is central to grasping the behavior of electric and magnetic fields in various environments.