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Electric energy is the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
 associated with the conservative
Conservative force

A conservative force is defined as a force with the following property: when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken....
 Coulomb forces between charged particle
Charged particle

In physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a Plasma , which is called the fourth state of matter because its properties are quite different from solids, liq...
s contained within a 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....
, where the reference potential energy is usually chosen to be zero for particles at infinite separation. It can be defined as the amount of work
Mechanical work

In physics, mechanical work is the amount of energy transferred by a force acting through a distance. Like energy, it is a scalar quantity, with SI of joules....
 one must apply to charged particles to bring them from infinite separation to some finite proximity configuration. This is also equal to the negative of the work of the Coulomb forces that the particles exert on each other during the quasistatic
Quasistatic process

In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. In practice, such processes can be approximated by performing them "very slowly"....
 move:

where



Sometimes people refer to the potential energy of a charge in an electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
.






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Electric energy is the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
 associated with the conservative
Conservative force

A conservative force is defined as a force with the following property: when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken....
 Coulomb forces between charged particle
Charged particle

In physics, a charged particle is a particle with an electric charge. It may be either a subatomic particle or an ion. A collection of charged particles, or even a gas containing a proportion of charged particles, is called a Plasma , which is called the fourth state of matter because its properties are quite different from solids, liq...
s contained within a 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....
, where the reference potential energy is usually chosen to be zero for particles at infinite separation. It can be defined as the amount of work
Mechanical work

In physics, mechanical work is the amount of energy transferred by a force acting through a distance. Like energy, it is a scalar quantity, with SI of joules....
 one must apply to charged particles to bring them from infinite separation to some finite proximity configuration. This is also equal to the negative of the work of the Coulomb forces that the particles exert on each other during the quasistatic
Quasistatic process

In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. In practice, such processes can be approximated by performing them "very slowly"....
 move:

where

  • is the work required to bring the system to a certain finite proximity configuration. "app" stands for applied, because this is work that must be applied to the system (or be supplied by another form of energy contained by the system) to configure it
  • is the work done by electrostatic inter-particle Coulomb forces during the move from infinity.


Sometimes people refer to the potential energy of a charge in an electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
. This actually refers to the potential energy of the system containing the charge and the other particles that created the electric field.

Furthermore, to calculate the work required to bring a charged particle into the vicinity of other particles, it is sufficient to know only the field generated by the other particles and the charge of the particle being moved. The field of the moving particle and the individual charges of the other particles do not need to be known.

Finally, it must be stressed that, even though this article talks about moving particles, the Coulomb force law on which this discussion is based only holds in the case of electrostatic systems. Therefore, any movement would have to be a quasistatic process
Quasistatic process

In thermodynamics, a quasistatic process is a thermodynamic process that happens infinitely slowly. In practice, such processes can be approximated by performing them "very slowly"....
.

Potential energy stored in a configuration of discrete charges

The potential energy between two charges is equal to the potential energy of one charge in the electric field of the other. That is to say, if generates a scalar electric potential field , which is a function of position , then . Also, a similar development gives .

This can be generalized to give an expression for a group of N charges, at positions :

where refers to the electric field due to all particles 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.

Alternatively, the factor of one half may be dropped if the sum is only performed once per particle pair. This is done in the examples below to cut down on the math.

One charged particle

The electric potential energy of a system containing only one point charge is zero, as no energy is required to move the charge particle from infinity to its location.

Two charged particles

Consider bringing a second charge into position:

where
ke is Coulomb's constant
q1, q2 are the two charges
r is the distance between the two particles


The electric potential energy will be negative if the charges have opposite sign and positive if the charges have the same sign. This simply means that potential energy is lost by a system of opposite charges moving together, which can be explained as 'opposite charges attract'.

Three or more charged particles

For 3 or more point charges, the electric potential energy of the system may be calculated by bringing individual charges into position one after another, and taking the sum of energy required to bring the additional charge into position.

where

ke is Coulomb's constant
q1, q2, ..., are the charges
rmn is the distance between two particles, m and n (e.g. r12).


Potential energy of a continuous charge distribution

The previous equation can again be generalized to give an expression of the potential energy of a continuous charge distribution.



where:
is the charge density
Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line , surface, or volume. It is measured in coulombs per metre , square metre , or cubic metre , respectively....
 of the distribution.
is the electric potential
Electric potential

At a point in space, the electric potential is the potential energy per unit of electric charge that is associated with a static electric field....
 at position r.


Energy stored in an electric field

One may take the equation for the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
 of a continuous charge distribution and put it in terms of the electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
.

Since Gauss' law for electric 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. It is measured in coulombs per metre , square metre , or cubic metre , respectively....
     including dipole
    Dipole

    In physics, there are two kinds of dipoles :*An electric dipole is a separation of positive and negative charge. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some, usually small, distance....
     charges bound in a material
  • is the vacuum permittivity.


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?s 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....
 and taking the area to be at infinity where





So, the energy density, or energy per unit volume of the electric field
Electric field

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field ....
 is:



The last two equations show that the energy stored in an electric field should be positive. But it is not always so with elementary particles in quantum mechanics. As an example let us take the hydrogen atom, which consists of two bound elementary charged particles; a proton and an electron. Both of them produce an electric field. The energy stored in this field consists of the energy stored in the electric field of each particle and an interaction energy. In quantum mechanics the energy stored in the electric field of an elementary charged particle is always zero, since there is no self-action. From this follows that the total energy stored in the electric field of the regarded system is equal to the interaction energy of the two charged particles. As the two particles have charges of the opposite signs, this energy is negative.

See also

  • Electric potential
    Electric potential

    At a point in space, the electric potential is the potential energy per unit of electric charge that is associated with a static electric field....
  • Potential energy
    Potential energy

    Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....