Electrical Energy

''For discussion of the physics and mathematics associated with electrical energy see Electromagnetism .'' ''For discussion of application and distribution of electrical energy see both Electric Power and AC Power '' Electrical energy refers to the Energy due to the interaction of Electric Charge s with an Electric Field , and the energy stored in that field. UNITS The SI unit of electrical energy is the Joule . In the context of use of electrical energy for lighting, heating, motors, and other applications, larger units such as the Kilowatt-hour , equivalent to 3.6 million joules, are used. 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 $q_1$ generates an electric field ''E'', then the potential energy is equal to $q_2E$ This can be generalized to give an expression for a group of ''N'' charges, $q_i$ at positions $r_i$ : : $U = rac{1}{2}\sum_i^N q_iV_{r_i}$ ''Note:'' The factor of one half accounts for the 'double counting' of charges. 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. : $U = rac{1}{2}\int \limits_{ ext{all space}} ho(r)V(r)d^3r$ where: : $ho(r)$ 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 uniform charge distribution and put it in terms of the Electric Field . Since : $\mathbf{ abla}\cdot\mathbf{E} = rac{ ho}{\epsilon_o}$ where : $\epsilon_o$ is the Permittivity of the medium :E is the electric field vector. then, : $U = rac{1}{2}\int \limits_{ ext{all space}} ho(r)V(r)d^3r$ : $= rac{1}{2}\int \limits_{ ext{all space}} \epsilon_o(\mathbf{ abla}\cdot{\mathbf{E}})V(r)d^3r$ also so, now : $U = rac{\epsilon_o}{2}\int \limits_{ ext{all space}} \mathbf{ abla}\cdot(\mathbf{E}V) d^3r - rac{\epsilon_o}{2}\int \limits_{ ext{all space}} (\mathbf{ abla}V)\mathbf{E} d^3r$ using the Divergence Theorem and taking the area to be at infinity where $V(\infty) = 0$ : $U = rac{\epsilon_o}{2}\int V\mathbf{E}\cdot dA - rac{\epsilon_o}{2}\int \limits_{ ext{all space}} (-\mathbf{E})\cdot\mathbf{E} d^3r$
	: <math> U E	rac{1}{2} \epsilon_o \leftE ight^2</math>

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Electrical Energy