vision electricity: Electric potential energy

Electric potential energy (also known as "electrostatic potential energy") is a potential energy associated with the conservative Coulomb forces within a defined system of point charges. The term "electrostatic potential energy" is preferred here because it seems less likely to be misunderstood. 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. ^[1]^:§25-1 The electrostatic potential energy of the system (U_E), relative to this zero, is equal to the total 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 desired system configuration:

$U_{\mathrm{E}} = \; W \;.$

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 people refer 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.^[1]^:§25-1
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 do not need to be known. Nonetheless, in many circumstances it is mathematically easier to add up all the pairwise potential energies (as below).
It is important to understand that electrostatics is a 18th-19th-century theory of hypothetical entities called "point charges". Electrostatics is categorically not a complete theory of the charged physical particles that make up the physical world, and are subject to the Heisenberg uncertainty principle and other laws of quantum mechanics.

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 one charge in the electrostatic potential generated by the other. That is to say, if charge

q 1

generates an electrostatic potential $\Phi_1(\mathbf r)$ , which is a function of position $\mathbf r$ , then $U_E = q_2 \Phi_1(\mathbf r_2)$ . Also, a similar development gives $U_E = q_1 \Phi_2(\mathbf r_1)$ .
This can be generalized to give an expression for a group of N charges,

q i

at positions $\mathbf r_i$ :

$U_E = \frac{1}{2}\sum_i^N q_i \Phi(\mathbf r_i)$

where, for each i value, $\Phi(\mathbf r_i)$ is the electrostatic potential due to all point charges except the one at $\mathbf r_i$
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 charge pair. This is done in the examples below to cut down on the math.

One point charge

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

Two point charges

Consider bringing a second point charge into position. The electrostatic potential

Φ(r 1)

due to charge 1 is

$\Phi(r_1) = \; k_{\mathrm{e}} q_1/r_1$

where k_e 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

$k_{\mathrm{e}} = \; 1/4 \pi \epsilon_0$ ,

where ε₀ is the electric constant. Hence we obtain:

$U_\mathrm{E} = \; \frac{q_1 q_2}{4 \pi \epsilon_0 r}$

where

q₁, q₂ are the two charges

r is the distance between the two charges

The electrostatic potential energy will be 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 charges; positive mutual potential energy to repulsion between two charges.

Three or more point charges

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

$U_\mathrm{E} = \frac{1}{4 \pi \epsilon_0} \left(\frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} + \cdots \right)$

where

q₁, q₂, ..., are the point charges

r_mn is the distance between two point charges, m and n (e.g., r₁₂).

Energy stored in an electrostatic field distribution

One may take the equation for the electrostatic potential energy of a continuous charge distribution and put it in terms of the electrostatic field.
Since Gauss' law for electrostatic field in differential form states

$\mathbf{\nabla}\cdot\mathbf{E} = \frac{\rho}{\epsilon_0}$

where

$\mathbf{E} \$ is the electric field vector
$\rho \$ is the total charge density including dipole charges bound in a material.

then,

$U = \frac{1}{2}\int \limits_{\text{all space}} \rho(r) \Phi(r)d^3r$

$= \frac{1}{2}\int \limits_{\text{all space}} \epsilon_0(\mathbf{\nabla}\cdot{\mathbf{E}})\Phi(r)d^3r$

so, now using the following divergence vector identity

$\nabla\cdot(\vec{A}{B}) = (\nabla\cdot\vec{A}){B} + \vec{A}\cdot(\nabla{B}) \Rightarrow (\nabla\cdot\vec{A}){B} = \nabla\cdot(\vec{A}{B}) - \vec{A}\cdot(\nabla{B})$

we have

$U = \frac{\epsilon_0}{2}\int \limits_{\text{all space}} \mathbf{\nabla}\cdot(\mathbf{E}\Phi) d^3r - \frac{\epsilon_0}{2}\int \limits_{\text{all space}} (\mathbf{\nabla}\Phi)\cdot\mathbf{E} d^3r$