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Electrical energy can refer to several closely related things. It can mean:

• the energy stored in an electric field
• the potential energy of a charged particle in an electric field
• the energy provided by electricity

In any of these cases, the SI unit of electrical energy is the joule. The unit used by many electrical utility companies is the watt-hour (Wh) or the kilowatt-hour (kWh).

Contents 1 Potential energy of a Charged Particle 2 Potential Energy stored in a configuration of charges 3 Potential Energy of a uniform charge distribution 4 Energy stored in an electric field 5 Energy of electricity 5.1 Usage note

[edit] Potential energy of a Charged Particle

Electrical energy is related to the position of an electric charge in an electric field. The electric potential energy of a charge q situated at an electric potential V(r) is equal to the product qV(r). The work needed to move this charge through a potential difference is given by the following equation:

<math>W_{ab} = qV_{ab} = q(V(b)-V(a))</math>

[edit] Potential Energy stored in a configuration of 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 <math>q_1</math> generates an electric field E, then the potential energy is equal to <math>q_2E</math>

This can be generalized to give an expression for a group of N charges, <math>q_i</math> at positions <math>r_i</math>:

<math>U = \frac{1}{2}\sum_i^N q_iV(r_i)</math>

Note: The factor of one half accounts for the 'double counting' of charges.

[edit] Potential Energy of a uniform charge distribution

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

<math>U = \frac{1}{2}\int_{All Space} \rho(r)V(r)d^3r </math>

where:

<math>\rho(r)</math> is the charge density of the distribution.

<math>V(r)</math> is the electric potential at position r.

[edit] Energy stored in an electric field

One may take the equation for the potential energy of a uniform charge distribution and put it in terms of the electric field.

Since

<math>\mathbf{\nabla}\cdot\mathbf{E} = \frac{\rho}{\epsilon_o}</math>

where <math>\epsilon_o</math> is the permittivity of the medium and E is the electric field vector.

then,

<math>U = \frac{1}{2}\int_{All Space} \rho(r)V(r)d^3r </math> (other math =) fun stuff at eceplydia.ca

<math> = \frac{1}{2}\int_{All Space} \epsilon_o(\mathbf{\nabla}\cdot{\mathbf{E}})V(r)d^3r </math>

also

<math>\mathbf{\nabla}\cdot(\mathbf{E}V) = (\mathbf{\nabla}V)\mathbf{E} + V(\mathbf{\nabla}\cdot\mathbf{E})</math>

so, now

<math> U = \frac{\epsilon_o}{2}\int_{All Space} \mathbf{\nabla}\cdot(\mathbf{E}V) d^3r - \frac{\epsilon_o}{2}\int_{All Space} (\mathbf{\nabla}V)\mathbf{E} d^3r</math>

using the divergence theorem and taking the area to be at infinity where <math>V(\infty) = 0</math>

<math> U = \frac{\epsilon_o}{2}\int V\mathbf{E}\cdot dA - \frac{\epsilon_o}{2}\int_{All Space} (-\mathbf{E})\cdot\mathbf{E} d^3r </math>

<math> = \int_{All Space} \frac{1}{2}\epsilon_o\left|{\mathbf{E}}\right|^2 d^3r</math>

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

<math> \eta = \frac{1}{2} \epsilon_o \left|E\right|^2</math>

[edit] Energy of electricity

Electrical energy is the amount of work that can be done by electricity.

[edit] Usage note

Frequently, the terms electrical energy and electric power are used interchangeably. However, in physics, and electrical engineering, "energy" and "power" have different meanings. Power is energy per unit time. In other words, the phrases "flow of power," and "consume a quantity of electric power" are both incorrect and should be changed to "flow of energy" and "consume a quantity of electrical energy."

ar:طاقة كهربائية cs:Elektrická energie da:Elektrisk energi fr:Énergie électrique hr:Električna energija io:Elektrika energio it:Energia elettrica he:אנרגיה חשמלית ja:電力量 pl:Energia elektryczna ru:Электрическая энергия sk:Elektrická energia sv:Elektrisk energi tr:Elektrik enerjisi zh:电能