Electrical energy

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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), which is the amount of energy used by a one-watt load, such as a tiny light bulb, drawing power for one hour. The kilowatt-hour (kWh), which is 1,000 times larger than a watt-hour, is a useful size for measuring the energy use of households and small businesses and also for the production of energy by small power plants. A typical house uses several hundred kilowatt-hours per month. The megawatt-hour (MWh), which is 1,000 times larger than the kilowatt-hour, is used for measuring the energy output of large power plants.

The terms "electrical energy" and "electric power" are frequently used interchangeably. However, in physics, and electrical engineering, "energy" and "power" have different meanings. Power is energy per unit time. The SI unit of power and electricity is the watt. One watt is a joule per second. 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."

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 See also

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:

$W_{ab} = qV_{ab} = q(V_{b}-V_{a}) \,$

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 $q 1$ generates an electric field E, then the potential energy is equal to $q 2 E$

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

$U = \frac{1}{2}\sum_i^N q_iV_{r_i}$

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

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

$U = \frac{1}{2}\int_{All Space} \rho(r)V(r)d^3r$

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

Since

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

where

ε o

is the permittivity of the medium

E is the electric field vector.

then,

$U = \frac{1}{2}\int_{All Space} \rho(r)V(r)d^3r$

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

also

$\mathbf{\nabla}\cdot(\mathbf{E}V) = (\mathbf{\nabla}V)\mathbf{E} + V(\mathbf{\nabla}\cdot\mathbf{E})$

so, now

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