Electric Field Energy: Meaning & Examples

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Lightning is arguably one of the most spectacular natural phenomena on Earth. A large amount of charge streaks its way through the sky and can be devastating to any living creature that gets in its path to the ground. This is due to the immense energy carried by a bolt of lightning - as much one billion joules! Apart from the blue flash that accompanies a lightning bolt, something else invisible carries this large amount of energy; the electric field. This energy is known as electric field energy.

Electric Field Energy Meaning

We know that electric fields exist due to the presence of charges, but they also carry energy that can be transformed, or used to do work. This is called electric field energy or electric potential energy and can be defined simply as follows:

The electric potential energy is the energy required to move a charge through an electric field.

The energy required to move an object is also known as the work done. We can hence think of this energy as the ability of a charged object to do work on another charged object. The figure below shows the electric field of a positive charge, represented by field lines pointing radially outward, interacting with another positive charge. The positive charge on the right-hand side experiences a force and hence moves to the right with acceleration $a$ . Work is done by the left-hand charge on the right-hand charge by applying an electric force on it and causing it to move.

Energy of the electrical field Work done on a positive charge StudySmarter The positive charge on the left exerts an electrostatic force on the charge on the right due to its electric field. The work done on the charge on the right is equal to the initial electric potential energy of the charge on the left, StudySmarter Originals

Electric Field Energy Calculation

We need to now write an expression for the electric potential energy and to do this we will consider the energy between two point charges as shown in the figure below. The electric potential energy $E_{p}$ between two point charges, one with charge $Q$ and the other with charge $q$ , separated by a distance $r$ is given by the equation

$E_{p} = \frac{q Q}{4 π ε_{0} r}$

where $ε_{0} = 8.85 \times 10^{- 12} {Fm}^{- 1}$ is the permittivity of free space, which is a constant. The charges are measured in units of Coulombs $(C)$ , and $r$ is given in meters $(m)$ . The unit for electric potential energy is the joule $(J)$ . As shown in the diagram below, it is clear that there is an inverse relationship between the energy and the separation distance, that is:

$E_{p} \propto \frac{1}{r}$

Energy of the electrical field Electrical potential energy between point charges StudySmarter The electric potential energy between two point charges is inversely proportional to their separation distance, StudySmarter Originals

Electric Field Energy in Capacitors

Capacitors are devices that can store electric potential energy and release it as charge through an electric circuit. They consist of parallel plates, and when charged will have a positive plate and a negative plate. We have shown the formula that is used to find the energy between two point charges, but we need to write one for the energy stored in a capacitor.

Let's assume that a capacitor has charge $Q$ stored on one of its plates, and a potential difference of $V$ between the plates. The electric potential energy stored in the capacitor is:

$E_{p} = \frac{1}{2} Q V$

A diagram of this scenario is shown below:

Energy of the electrical field Energy stored in a capacitor StudySmarter The energy stored in a capacitor is half the product of the charge on one of the plates and the potential difference between the plates, StudySmarter Originals

Electric Field Energy Density Derivation

The electric field energy changes with distance as we've seen before, so it makes sense to think of the energy of an electric field within a certain region of space. The energy per unit volume is called the electric field energy density $\frac{E_{p}}{V}$ and we can derive an equation for this quantity as follows:

$E_{p} = \frac{1}{2} Q V_{0} = \frac{1}{2} C {V_{0}}^{2} = \frac{1}{2} (\frac{ε_{0} A}{d}) {(E d)}^{2} = \frac{1}{2} ε_{0} E^{2} (A d) = \frac{1}{2} ε_{0} E^{2} V \frac{E_{p}}{V} = \frac{1}{2} ε_{0} E^{2}$

For an electric field strength $E$ with a potential difference between the plates $V_{0}$ and volume $V = A d$ . We've also used the following expressions in the derivation above.

$C = \frac{Q}{V}$ , capacitance is the amount of charge a capacitor stores per unit of potential difference.
$C = \frac{ε_{0} A}{d}$ , the capacitance of a capacitor depends on the area of its plates $A$ and their separation $d$ .
$E = \frac{V_{0}}{d}$ , the electric field strength between the plates of a capacitor is the potential difference between the plates divided by their separation.

Note that some knowledge of capacitors is required for the above derivation. The volume between the two parallel plates of the capacitor is depicted graphically below.

Energy of the electrical field Volume between plates of a capacitor StudySmarter The volume between the plates of a capacitor is the product of the surface area of the face of one plate and the separation between the plates, StudySmarter Originals

The energy density is simply the amount of energy per unit volume contained within this region between the plates.

Electric Field Energy Example

We can test our understanding of electric field energy by considering the example below.

Q. What is the electric potential energy between two identical charges of $1.60 \times 10^{- 19} C$ separated by a distance of $2.0 \times 10^{- 4} m$ ?

A. For this problem, we can see that $Q = q = 1.60 \times 10^{- 19} C$ , and the distance between the charges $r = 2.0 \times 10^{- 4} m$ . We can use these values in the equation for electric potential energy as follows:

$E_{p} = \frac{q Q}{4 π ε_{0} r} = \frac{{(1.60 \times 10^{- 19} C)}^{2}}{4 π (8.85 \times 10^{- 12} {Fm}^{- 1}) (2.0 \times 10^{- 4} m)} = 1.2 \times 10^{- 24} J$

The electric potential energy that exists between the two charges is, therefore $1.2 \times 10^{- 24} J$ (not much!).

Electric Field Energy - Key takeaways

The electric field energy or electric potential energy is the energy required to move a charge through an electric field.
It is the work done by a charged object in moving another charged object.
The electric potential energy $E_{p}$ between two point charges, one with charge $Q$ and the other with charge $q$ , separated by a distance $r$ is given by the equation:
$E_{p} = \frac{q Q}{4 π ε_{0} r}$

There is an inverse relationship between the energy and the separation distance, that is:
$E_{p} \propto \frac{1}{r}$

The electric potential energy stored in a capacitor that has charge $Q$ stored on one of its plates, and a potential difference of $V$ between the plates is:
$E_{p} = \frac{1}{2} Q V$
For an electric field strength $E$ in a region of volume $V$ ,the electric field energy density $\frac{E_{p}}{V}$ is given by:

$\frac{E_{p}}{V} = \frac{1}{2} ε_{0} E^{2}$

The energy density for a capacitor is the amount of energy per unit volume contained within the region between the plates.

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The electric field energy or electric potential energy is the energy required to move a ... through an electric field.

charge

There is an inverse-square relationship between the energy and the separation distance.

False

The electric field energy is the energy required to move a charge through a gravitational field.

False

The electric field energy is equal to the potential difference between two points.

False

As the distance between two point charges increases, the electric potential energy between them ...

decreases

The electric potential energy stored in a capacitor's electric field is directly proportional to the potential difference between the plates when charge is constant.

True

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