Electric potential energy  

Common symbols  U_{E} 
SI unit  joule (J) 
Derivations from other quantities  U_{E} = C · V^{2} / 2 
Electric potential energy, is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged objects.
The term "electric potential energy" is used to describe the potential energy in systems with timevariant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with timeinvariant electric fields.
The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration.
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The electrostatic potential energy can also be defined from the electric potential as follows:
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The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule). In the CGS system the erg is the unit of energy, being equal to 10^{7} Joules. Also electronvolts may be used, 1 eV = 1.602×10^{19} Joules.
The electrostatic potential energy, U_{E}, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is:
,
where is Coulomb's constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the chargesi.e., an electron would have a negative value of charge when placed in the formula). The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula.
The electrostatic potential energy, U_{E}, of one point charge q in the presence of n point charges Q_{i}, taking an infinite separation between the charges as the reference position, is:
,
where is Coulomb's constant, r_{i} is the distance between the point charges q and Q_{i}, and q and Q_{i} are the assigned values of the charges.
The electrostatic potential energy U_{E} stored in a system of N charges q_{1}, q_{2}, ..., q_{N} at positions r_{1}, r_{2}, ..., r_{N} respectively, is:

where, for each i value, ?(r_{i}) is the electrostatic potential due to all point charges except the one at r_{i},^{[note 2]} and is equal to:
,
where r_{ij} is the distance between q_{i} and q_{j}.
The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic force against which an external agent must do work in moving the point charge from infinity to its final location.
A common question arises concerning the interaction of a point charge with its own electrostatic potential. Since this interaction doesn't act to move the point charge itself, it doesn't contribute to the stored energy of the system.
Consider bringing a point charge, q, into its final position near a point charge, Q_{1}. The electrostatic potential ?(r) due to Q_{1} is
Hence we obtain, the electric potential energy of q in the potential of Q_{1} as
where r_{1} is the separation between the two point charges.
The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q_{1} due to two charges Q_{2} and Q_{3}, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q_{2} and Q_{3}.
The electrostatic potential energy stored in the system of three charges is:
The energy density, or energy per unit volume, , of the electrostatic field of a continuous charge distribution is:
Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat. This is known as the Joule effect. A capacitor stores it in its electric field. The total electrostatic potential energy stored in a capacitor is given by
where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.
The total electrostatic potential energy may also be expressed in terms of the electric field in the form
where is the electric displacement field within a dielectric material and integration is over the entire volume of the dielectric.
The total electrostatic potential energy stored within a charged dielectric may also be expressed in terms of a continuous volume charge, ,
where integration is over the entire volume of the dielectric.
These latter two expressions are valid only for cases when the smallest increment of charge is zero () such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.