Electric energy is the potential energy associated with the conservative Coulomb forces between charged particles contained within a 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 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 move:
UE = Wapp = -W∞
Where
Wapp 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.
W∞ 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. 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.
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 q1 generates a scalar electric potential field V1(r), which is a function of position r, then UE = q2V1(r2). Also, a similar development gives UE = q2V1(r2).
This can be generalized to give an expression for a group of N charges, qi at positions r1:
UE = ½ ∑ qi V (ri)
Where V (ri) refers to the electric field due to all particles except the one at ri
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:
UE = ke (q1q2/r)
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.
UE = ke (q1q2/r12 + q1q2/r13 + q1q2/r23 + …)
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.
U = 1/2 ∫ p(r)V(r)d3r
where:
ρ(r) is the charge density of the distribution.
V(r) is the electric potential at position r.
Energy stored in an electric field
One may take the equation for the potential energy of a continuous charge distribution and put it in terms of the electric field.
Since Gauss' law for electric field in differential form states
▼.E = ρ/є0
where
E is the electric field vector
ρ is the total charge density including dipole charges bound in a material
є0 is the vacuum permittivity.
then,
U = 1/2 ∫ ρ(r)V(r)d3r
= 1/2 ∫ є0(▼.E)V(r) d3r
so, now using the following divergence vector identity
▼.(AB) = (▼. A)B+A . ( ▼B) → (▼. A)B = ▼. (AB) – A . (▼B)
we have
U = є0/2 ∫ ▼. (EV)d3r - є0/2 ∫ (▼V) . Ed3r
using the divergence theorem and taking the area to be at infinity where V(∞) = 0
U = є0/2 ∫ VE . dA - є0/2 ∫ (-E) . Ed3r
= ∫ 1/2 є0 |E|2 d3r
So, the energy density, or energy per unit volume of the electric field is:
ue = 1/2 є0E2
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 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.
References:
- http://en.wikipedia.org
- Halliday, David; Resnick, Robert; Walker, Jearl (1997). "Electric Potential" (in English). Fundamentals of Physics (5th ed.). John Wiley & Sons. ISBN 0-471-10559-7.
