Electric potential energy
Encyclopedia
Electric potential energy, or electrostatic potential energy, is a potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

(measured in joules
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

) that results from conservative
Conservative force
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.It is possible to define a numerical value of...

Coulomb forces and is associated with the configuration of a particular set of point charges
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

within a defined system
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...

. Not to be confused with the term electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

(measured in volts
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

), the term "electric potential energy" is used to describe the potential energy in systems with electric fields
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

that change with time (time variant), while the term "electrostatic potential energy" is used to describe the potential energy in systems with electric fields
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

that do not change with time (time invariant).

The reference zero is usually taken to be a state in which the individual point charges are very well separated ("are at infinite separation") and are at rest. The electric potential energy of a system (UE), relative to this zero, is equal to the total work
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...

'W that must be done by a hypothetical external agent in order to bring the charges slowly, one by one, from infinite separation, to the stated system configuration.

In this process the external agent is deemed to provide or absorb any relevant work, and the point charge being slowly moved gains no kinetic energy.

Sometimes reference is made to the potential energy of a charge in an electrostatic field. This actually refers to the potential energy of the system containing the charge and the other charges that created the electrostatic field.

Electrical energy is energy newly derived from electrical potential energy. When loosely used to describe energy absorbed or delivered by an electrical circuit (for example, one provided by an electric power
Electric power
Electric power is the rate at which electric energy is transferred by an electric circuit. The SI unit of power is the watt.-Circuits:Electric power, like mechanical power, is represented by the letter P in electrical equations...

utility) "electrical energy" refers to energy which has been converted from electrical potential energy. This energy is supplied by the combination of electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

and electrical potential that is delivered by the circuit. At the point that this electrical potential energy has been converted to another type of energy, it ceases to be electrical potential energy. Thus, all electrical energy is potential energy before it is delivered to the end-use. Once converted from potential energy, electrical energy can always be described as another type of energy (heat, light, motion, etc.).

To calculate the work required to bring a point charge into the vicinity of other (stationary) point charges, it is sufficient to know only (a) the total field generated by the other charges and (b) the charge of the point charge being moved. The field due to the charge being moved and the values of the other charges are not required. Nonetheless, in many circumstances it is mathematically easier to add up all the pairwise potential energies (as below).

## Electrostatic potential energy stored in a configuration of discrete point charges

The mutual electrostatic potential energy of two charges is equal to the potential energy of a charge in the electrostatic potential generated by the other. That is to say, if charge generates an electrostatic potential , which is a function of position , then . Doing the same calculation with respect to the other charge, we obtain .
This can be generalized to give an expression for a group of N charges , ,..., at positions , ,..., , respectively:

where, for each i value, is the electrostatic potential due to all point charges except the one at

Note: The factor of one half accounts for the 'double counting' of charge pairs. For example, consider the case of just two charges.

#### One point charge

The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic potential against which an external agent must do work in moving the point charge from infinity to its final location. One should carefully consider the possibility of the point charge interacting with its own electrostatic potential. However, since such a potential at the location of the point charge itself is infinite, this "self-energy" is intentionally excluded from an evaluation of the total (finite) electrostatic potential energy of the system. Moreover, one may argue that since the electrostatic potential due to the point charge itself provides no work in moving the point charge around, this interaction is unimportant for most purposes.

#### Two point charges

Consider bringing a second point charge, , into its final position in the vicinity of the first point charge, . The electrostatic potential due to is

where ke is Coulomb's constant. In the International System of Quantities, which has been the preferred international system since the 1970s and forms the basis for the definition of SI units, the Coulomb constant is given by
,

where is the electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...

. Hence we obtain, potential energy of q in the potential of Q as

where is the distance between the two point charges.

The electrostatic potential energy is negative if the charges have opposite sign and positive if the charges have the same sign. Negative mutual potential energy corresponds to attraction between two point charges; positive mutual potential energy to repulsion between two point charges.

### Three or more point charges

The electrostatic potential energy of q due to two charges and should not be confused with the electrostatic potential energy of the system of three charges, because the latter also includes the electrostatic potential energy of the system of the two charges and .

For three or more point charges, the electrostatic potential energy of the system may be calculated by the total amount of work done by an external agent in bringing individual point charges into their final positions one after another.

NOTE
Here, is the relative permittivity of free space. When the charge is in a medium other than free space / air ,the relative permittivity, , has to be taken into account where k is the dielectric constant of the medium. K is the ratio of the electrostatic force on the charges in free space to the electrostatic force on the charges in the respective medium to its energy

## Energy stored in an electrostatic field distribution

One may take the equation for the electrostatic potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

of a continuous charge distribution and put it in terms of the electrostatic field.

Since Gauss' law for electrostatic field in differential form states

where
• is the electric field vector
• is the total charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

including dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

charges bound in a material,

then,

so, now using the following divergence vector identity

we have

using the divergence theorem
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...

and taking the area to be at infinity where

So, the energy density, or energy per unit volume of the electrostatic field is:

## Energy in electronic elements

Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat, and a capacitor stores it in its electric field.

The total electric potential energy stored in a capacitor is given by

where is the capacitance , is the electric potential difference, and the charge stored in the capacitor.