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Electric potential energy, or Electrostatic 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 time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields.

Definition

The electric potential energy of a system of point charges is defined as the work required assembling this system of charges by bringing them close together, as in the system from an infinite distance.

The electrostatic potential energy, UE, of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position rref[note 1] to that position r.:§25–1[note 2]

$$U_{\mathrm {E} }(\mathbf {r} )=-W_{r_{\rm {ref}}\rightarrow r}=-\int _{{\mathbf {r} }_{\rm {ref}}}^{\mathbf {r} }q\mathbf {E} (\mathbf {r'} )\cdot \mathrm {d} \mathbf {r'} },$$

where E is the electrostatic field and dr' is the displacement vector in a curve from the reference position rref to the final position r.

The electrostatic potential energy can also be defined from the electric potential as follows:

The electrostatic potential energy, UE, of one point charge q at position r in the presence of an electric potential Φ {\displaystyle \scriptstyle \Phi } \scriptstyle \Phi is defined as the product of the charge and the electric potential.

$$U_{{\mathrm {E}}}({\mathbf r})=q\Phi ({\mathbf r}),$$

where $$\Phi$$ is the electric potential generated by the charges, which is a function of position r.

Units

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 J. Also electronvolts may be used, 1 eV = 1.602×10−19 J.
Electrostatic potential energy of one point charge
One point charge q in the presence of another point charge Q
A point charge q in the electric field of another charge Q.

The electrostatic potential energy, UE, 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:

$$U_{E}(r)=k_{e}{\frac {qQ}{r}},$$

where $$k_{e}={\frac {1}{4\pi \varepsilon _{0}}}$$ is Coulomb's constant, r is the distance between the point charges q & Q, and q & Q are the charges (not the absolute values of the charges—i.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.

Outline of proof

One point charge q in the presence of n point charges Qi
Electrostatic potential energy of q due to Q1 and Q2 charge system: $$U_{E}=q{\frac {1}{4\pi \varepsilon _{0}}}\left({\frac {Q_{1}}{r_{1}}}+{\frac {Q_{2}}{r_{2}}}\right)$$

The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qi, taking an infinite separation between the charges as the reference position, is:

$$U_{E}(r)=k_{e}q\sum _{{i=1}}^{n}{\frac {Q_{i}}{r_{i}}},$$

where $$k_{e}={\frac {1}{4\pi \varepsilon _{0}}}$$ is Coulomb's constant, ri is the distance between the point charges q & Qi, and q & Qi are the signed values of the charges.
Electrostatic potential energy stored in a system of point charges

The electrostatic potential energy UE stored in a system of N charges q1, q2, ..., qN at positions r1, r2, ..., rN respectively, is:

$$U_{\mathrm {E} }={\frac {1}{2}}\sum _{i=1}^{N}q_{i}\Phi (\mathbf {r} _{i})={\frac {1}{2}}k_{e}\sum _{i=1}^{N}q_{i}\sum _{j=1}^{N(j\neq i)}{\frac {q_{j}}{r_{ij}}}},$$ (1)

where, for each i value, Φ(ri) is the electrostatic potential due to all point charges except the one at ri,[note 3] and is equal to:

$$\Phi (\mathbf {r} _{i})=k_{e}\sum _{j=1}^{N(j\neq i)}{\frac {q_{j}}{\mathbf {r} _{ij}}}},$$

where rij is the distance between qj and qi.

Outline of proof

Energy stored in a system of one point charge

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.
Energy stored in a system of two point charges

Consider bringing a point charge, q, into its final position near a point charge, Q1. The electrostatic potential Φ(r) due to Q1 is

$$\Phi (r)=k_{e}{\frac {Q_{1}}{r}}$$

Hence we obtain, the electric potential energy of q in the potential of Q1 as

$$U_{E}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {qQ_{1}}{r_{1}}}$$

where r1 is the separation between the two point charges.
Energy stored in a system of three point charges

The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q1 due to two charges Q2 and Q3, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q2 and Q3.

The electrostatic potential energy stored in the system of three charges is:

$$U_{\mathrm {E} }={\frac {1}{4\pi \varepsilon _{0}}}\left[{\frac {Q_{1}Q_{2}}{r_{12}}}+{\frac {Q_{1}Q_{3}}{r_{13}}}+{\frac {Q_{2}Q_{3}}{r_{23}}}\right]}$$

Outline of proof

Energy stored in an electrostatic field distribution

The energy density, or energy per unit volume, $${\frac {dU}{dV}}$$, of the electrostatic field of a continuous charge distribution is:

$$u_{e}={\frac {dU}{dV}}={\frac {1}{2}}\varepsilon _{0}\left|{{\mathbf {E}}}\right|^{2}.$$

Outline of proof

Energy stored in electronic elements
The electric potential energy stored in a capacitor is UE=½ CV2

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 electric potential energy stored in a capacitor is given by

$$U_{E}={\frac {1}{2}}QV={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}$$

where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.

Outline of proof

The total electrostatic potential energy may also be expressed in terms of the electric field in the form

$$U_{E}={\frac {1}{2}}\int _{V}\mathrm {E} \cdot \mathrm {D} dV}$$

where D {\displaystyle \mathrm {D} } \mathrm {D} 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, $$\rho$$ ,

$$U_{E}={\frac {1}{2}}\int _{V}\rho \Phi dV}$$

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 ( $$dq\to 0}$$ ) such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.
Notes

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.
Alternatively, it can also be defined as the work W done by an external force to bring it from the reference position rref to some position r. Nonetheless, both definitions yield the same results.

The factor of one half accounts for the 'double counting' of charge pairs. For example, consider the case of just two charges.

References

Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0-471-92712-0

Halliday, David; Resnick, Robert; Walker, Jearl (1997). "Electric Potential". Fundamentals of Physics (5th ed.). John Wiley & Sons. ISBN 0-471-10559-7.

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