# Electric potential energy

Last updated
Electric potential energy
Common symbols
UE
SI unit joule (J)
Derivations from
other quantities
UE = C · V 2 / 2

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.

In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.

The joule is a derived unit of energy in the International System of Units. It is equal to the energy transferred to an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre. It is also the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second. It is named after the English physicist James Prescott Joule (1818–1889).

A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the taken path. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.

## Contents

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.

A time-variant system is a system that is not time invariant (TIV). Roughly speaking, its output characteristics depend explicitly upon time. In other words, a system in which certain quantities governing the system's behavior change with time, so that the system will respond differently to the same input at different times.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain, then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".

## 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. [1] [2] :§25-1 [note 2]

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

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 ${\displaystyle \scriptstyle \Phi }$ is defined as the product of the charge and the electric potential.

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

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

## Units

The SI unit of electric potential energy is the 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.

James Prescott Joule was an English physicist, mathematician and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work. This led to the law of conservation of energy, which in turn led to the development of the first law of thermodynamics. The SI derived unit of energy, the joule, is named after him.

The erg is a unit of energy and work equal to 10−7 joules. It originated in the centimetre–gram–second (CGS) system of units. It has the symbol erg. The erg is not an SI unit. Its name is derived from ergon (ἔργον), a Greek word meaning work or task.

## Electrostatic potential energy of one point charge

### One point charge q in the presence of another point 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:

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

where ${\displaystyle k_{e}={\frac {1}{4\pi \varepsilon _{0}}}}$ 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.

The electron is a subatomic particle, symbol
e
or
β
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

Coulomb's law, or Coulomb's inverse-square law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force. The quantity of electrostatic force between stationary charges is always described by Coulomb's law. The law was first published in 1785 by French physicist Charles-Augustin de Coulomb, and was essential to the development of the theory of electromagnetism, maybe even its starting point, because it was now possible to discuss quantity of electric charge in a meaningful way.

### One point charge q in the presence of n point charges Qi

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:

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

where ${\displaystyle k_{e}={\frac {1}{4\pi \varepsilon _{0}}}}$ 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:

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

(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:

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

where rij is the distance between qj and qi.

### 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 potential 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

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

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

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

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:

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

## Energy stored in an electrostatic field distribution

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

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

## Energy stored in electronic elements

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

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

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

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

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

where ${\displaystyle \mathrm {D} }$ is the displacement of the electric 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, ${\displaystyle \rho }$,

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

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

## Notes

1. 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.
2. 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.
3. The factor of one half accounts for the 'double counting' of charge pairs. For example, consider the case of just two charges.

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## References

1. Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN   0-471-92712-0
2. Halliday, David; Resnick, Robert; Walker, Jearl (1997). "Electric Potential". Fundamentals of Physics (5th ed.). John Wiley & Sons. ISBN   0-471-10559-7.