View source for Electric potential energy

{{Distinguish|Electric potential|Electric power}}
{{About|the physical magnitude Electric Potential Energy|electrical energy|Electrical energy|energy sources|Energy development|electricity generation|Electricity generation}}
{{Infobox Physical quantity
|bgcolour={default}
|name=Electric potential energy
|image=
|caption=
|unit=[[joule]] (J)
|symbols=U<sub>E</sub>
|derivations=U<sub>E</sub> = [[Capacitance|C]] · [[Electric potential|V]]<sup>2</sup> / 2
}}
{{Electromagnetism|cTopic=Electrostatics}}

'''Electric potential energy''', or '''Electrostatic potential energy''', is a [[potential energy]] (measured in [[joule]]s) that results from [[conservative force|conservative]] [[Coulomb force]]s and is associated with the configuration of a particular set of point [[electric charge|charges]] within a defined [[physical system|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 system|time-variant]] [[electric field]]s, while the term "electrostatic potential energy" is used to describe the potential energy in systems with [[time-invariant system|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, ''U<sub>E</sub>'', of one [[point charge]] ''q'' at position '''r''' in the presence of an [[electric field]] '''E''' is defined as the negative of the [[Work (electrical)|work]] ''W'' done by the [[electrostatic force]] to bring it from the reference position '''r'''<sub>ref</sub><ref group="note">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.</ref> to that position '''r'''.<ref>Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 {{ISBN|0-471-92712-0}}</ref><ref name="HRW1997">{{cite book |last=Halliday |first=David |author2=Resnick, Robert |author3=Walker, Jearl  |title=Fundamentals of Physics |url=https://archive.org/details/fundamentalsofp000davi |url-access=registration |edition=5th |year=1997 |publisher=[[John Wiley & Sons]] |chapter=Electric Potential |isbn=0-471-10559-7}}</ref>{{rp|§25-1}}<ref group="note">Alternatively, it can also be defined as the [[Mechanical work|work]] ''W'' done by an external force to bring it from the reference position '''r'''<sub>ref</sub> to some position '''r'''. Nonetheless, both definitions yield the same results.</ref>

{{Equation box 1
|indent=:
|equation=<math>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'}</math>,
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

:where '''E''' is the electrostatic field and d'''r'''' is the displacement vector in a curve from the reference position '''r'''<sub>ref</sub> to the final position '''r'''.

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

:The electrostatic potential energy, ''U<sub>E</sub>'', of one point charge ''q'' at position '''r''' in the presence of an [[electric potential]] <math>\scriptstyle \Phi</math> is defined as the product of the charge and the electric potential.

{{Equation box 1
|indent=:
|equation=<math>U_\mathrm{E}(\mathbf r) = q \Phi(\mathbf r)</math>,
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

:where <math>\scriptstyle \Phi</math> 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|CGS system]] the [[erg]] is the unit of energy, being equal to 10<sup>−7</sup> J. Also [[electronvolts]] may be used, 1 eV = 1.602×10<sup>−19</sup> J.

==Electrostatic potential energy of one point charge==

===One point charge ''q'' in the presence of another point charge ''Q''===

[[File:Point Charge q in an electric field.svg|right|A point charge q in the electric field of another charge Q.|thumb|434px]]

The electrostatic potential energy, ''U<sub>E</sub>'', 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:

{{Equation box 1
|indent=:
|equation=<math> U_E(r) =  k_e\frac{qQ}{r}</math>,
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>k_e = \frac{1}{4\pi\varepsilon_0}</math> 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.

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Outline of proof
|-
|
The electrostatic force '''F''' acting on a charge ''q'' can be written in terms of the electric field '''E''' as
:<math> \mathbf{F} = q\mathbf{E} </math>,

By definition, the change in electrostatic potential energy, ''U<sub>E</sub>'', of a point charge ''q'' that has moved from the reference position '''r'''<sub>ref</sub> to position '''r''' in the presence of an electric field '''E''' is the negative of the work done by the [[electrostatic force]] to bring it from the reference position '''r'''<sub>ref</sub> to that position '''r'''.

:<math> U_E(r) - U_E(r_{\rm ref}) = -W_{r_{\rm ref} \rightarrow r } = -\int_{{r}_{\rm ref}}^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s} </math>.

where:
*'''r''' = position in 3d space of the charge ''q'', using cartesian coordinates '''r''' = (''x'', ''y'', ''z''), taking the position of the ''Q'' charge at '''r''' = (0,0,0), the scalar ''r'' = |'''r'''| is the [[Norm (mathematics)|norm]] of the position vector,
*d'''s''' = differential [[displacement vector]] along a path ''C'' going from '''r'''<sub>ref</sub> to '''r''',
*<math> \scriptstyle W_{r_{\rm ref} \rightarrow r } </math> is the work done by the electrostatic force to bring the charge from the reference position '''r'''<sub>ref</sub> to '''r''',

Usually ''U<sub>E</sub>'' is set to zero when '''r'''<sub>ref</sub> is infinity:

:<math> U_E (r_{\rm ref}=\infty) = 0 </math>

so

:<math> U_E(r) = - \int_\infty^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s} </math>

When the [[Curl (mathematics)|curl]] {{nowrap|'''&nabla;''' &times; '''E'''}} is zero, the line integral above does not depend on the specific path ''C'' chosen but only on its endpoints. This happens in time-invariant electric fields. When talking about electrostatic potential energy, time-invariant electric fields are always assumed so, in this case, the electric field is [[conservative vector field|conservative]] and Coulomb's law can be used.

Using [[Coulomb's law]], it is known that the electrostatic force '''F''' and the electric field '''E ''' created by a discrete point charge ''Q'' are radially directed from ''Q''. By the definition of the position vector '''r''' and the displacement vector '''s''', it follows that '''r''' and '''s''' are also radially directed from ''Q''. So, '''E''' and d'''s''' must be parallel:

:<math> \mathbf{E} \cdot \mathrm{d} \mathbf{s} = |\mathbf{E}| \cdot |\mathrm{d}\mathbf{s}|\cos(0) = E \mathrm{d}s </math>

Using Coulomb's law, the electric field is given by

:<math> |\mathbf{E}| = E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{s^2} </math>

and the integral can be easily evaluated:

:<math> U_E(r) = -\int_\infty^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s} = -\int_\infty^r \frac{1}{4\pi\varepsilon_0}\frac{qQ}{s^2}{\rm d}s = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{r} = k_e\frac{qQ}{r} </math>

|}

===One point charge ''q'' in the presence of ''n'' point charges ''Q<sub>i</sub>''===
[[File:Electric potential energy 3 charge.gif|thumb|Electrostatic potential energy of ''q'' due to ''Q''<sub>1</sub> and ''Q''<sub>2</sub> charge system:<math>U_E = q\frac{1}{4 \pi \varepsilon_0} \left(\frac{Q_1}{r_1} + \frac{Q_2}{r_2} \right) </math>]]
The electrostatic potential energy, ''U<sub>E</sub>'', of one point charge ''q'' in the presence of ''n'' point charges ''Q<sub>i</sub>'', taking an infinite separation between the charges as the reference position, is:

{{Equation box 1
|indent=:
|equation=<math> U_E(r) =  k_e q \sum_{i=1}^n \frac{Q_i}{r_i}</math>,
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

where <math>k_e = \frac{1}{4\pi\varepsilon_0}</math> is [[Coulomb's constant]], ''r<sub>i</sub>'' is the distance between the point charges ''q'' & ''Q<sub>i</sub>'', and ''q'' & ''Q<sub>i</sub>'' are the signed values of the charges.

==Electrostatic potential energy stored in a system of point charges==
The electrostatic potential energy ''U''<sub>E</sub> stored in a system of ''N'' charges ''q''<sub>1</sub>, ''q''<sub>2</sub>, ..., ''q<sub>N</sub>'' at positions '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub> respectively, is:
{{NumBlk||

{{Equation box 1
|indent=:
|equation=<math>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\ne i)} \frac{q_j}{r_{ij}}</math>,
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

|{{EquationRef|1}}}}

where, for each ''i'' value, Φ('''r'''<sub>''i''</sub>) is the electrostatic potential due to all point charges except the one at '''r'''<sub>''i''</sub>,<ref group="note">The factor of one half accounts for the 'double counting' of charge pairs. For example, consider the case of just two charges.</ref> and is equal to:

<math>\Phi(\mathbf{r}_i) = k_e\sum_{j=1}^{N(j\ne i)} \frac{q_j}{\mathbf{r}_{ij}}</math>,

where ''r''<sub>ij</sub> is the distance between q<sub>j</sub> and q<sub>i</sub>.

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Outline of proof
|-
|
The electrostatic potential energy ''U''<sub>E</sub> stored in a system of two charges is equal to the electrostatic potential energy of a charge in the [[electrostatic potential]] generated by the other. That is to say, if charge q<sub>1</sub> generates an electrostatic potential Φ<sub>1</sub>, which is a function of position '''r''', then

:<math>U_\mathrm{E} = q_2 \Phi_1(\mathbf r_2).</math>

Doing the same calculation with respect to the other charge, we obtain

:<math>U_\mathrm{E} = q_1 \Phi_2(\mathbf r_1).</math>

The electrostatic potential energy is mutually shared by <math>q_1</math> and <math>q_2</math>, so the total stored energy is

: <math>U_E = \frac{1}{2}\left[q_2 \Phi_1(\mathbf r_2) + q_1 \Phi_2(\mathbf r_1)\right]</math>

This can be generalized to say that the electrostatic potential energy ''U''<sub>E</sub> stored in a system of ''N'' charges ''q''<sub>1</sub>, ''q''<sub>2</sub>, ..., ''q<sub>N</sub>'' at positions '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, ..., '''r'''<sub>''N''</sub> respectively, is:

<math>U_\mathrm{E} = \frac{1}{2}\sum_{i=1}^N q_i \Phi(\mathbf{r}_i)</math>.

|}

===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, ''Q''<sub>1</sub>. The electrostatic potential Φ('''r''') due to ''Q''<sub>1</sub> is

:<math> \Phi(r) = k_e \frac{Q_1}{r} </math>

Hence we obtain, the electric potential energy of ''q'' in the potential of ''Q''<sub>1</sub> as

:<math>U_E = \frac{1}{4\pi\varepsilon_0} \frac{q Q_1}{ r_1 }</math>

where r<sub>1</sub> 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 ''Q''<sub>1</sub> due to two charges ''Q''<sub>2</sub> and ''Q''<sub>3</sub>, because the latter doesn't include the electrostatic potential energy of the system of the two charges ''Q''<sub>2</sub> and ''Q''<sub>3</sub>.

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

:<math>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]</math>

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Outline of proof
|-
|
Using the formula given in ({{EquationNote|1}}), the electrostatic potential energy of the system of the three charges will then be:

:<math>U_\mathrm{E} = \frac{1}{2} \left[ Q_1 \Phi(\mathbf{r}_1) + Q_2 \Phi(\mathbf{r}_2) + Q_3 \Phi(\mathbf{r}_3) \right]</math>

Where <math>\Phi(\mathbf{r}_1)</math> is the electric potential in '''r'''<sub>1</sub> created by charges ''Q''<sub>2</sub> and ''Q''<sub>3</sub>, <math>\Phi(\mathbf{r}_2)</math> is the electric potential in '''r'''<sub>2</sub> created by charges ''Q''<sub>1</sub> and ''Q''<sub>3</sub>, and <math>\Phi(\mathbf{r}_3)</math> is the electric potential in '''r'''<sub>3</sub> created by charges ''Q''<sub>1</sub> and ''Q''<sub>2</sub>. The potentials are:

:<math>\Phi(\mathbf{r}_1) = \Phi_2(\mathbf{r}_1) + \Phi_3(\mathbf{r}_1) = \frac{1}{4\pi\varepsilon_0} \frac{Q_2}{r_{12}} + \frac{1}{4\pi\varepsilon_0} \frac{Q_3}{r_{13}}</math>
:<math>\Phi(\mathbf{r}_2) = \Phi_1(\mathbf{r}_2) + \Phi_3(\mathbf{r}_2) = \frac{1}{4\pi\varepsilon_0} \frac{Q_1}{r_{21}} + \frac{1}{4\pi\varepsilon_0} \frac{Q_3}{r_{23}}</math>
:<math>\Phi(\mathbf{r}_3) = \Phi_1(\mathbf{r}_3) + \Phi_2(\mathbf{r}_3) = \frac{1}{4\pi\varepsilon_0} \frac{Q_1}{r_{31}} + \frac{1}{4\pi\varepsilon_0} \frac{Q_2}{r_{32}}</math>

Where ''r''<sub>ab</sub> is the distance between charge ''Q''<sub>a</sub> and ''Q''<sub>b</sub>.

If we add everything:

:<math>U_\mathrm{E} = \frac{1}{2} \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_1}{r_{21}} + \frac{Q_2 Q_3}{r_{23}} + \frac{Q_3 Q_1}{r_{31}} +  \frac{Q_3 Q_2}{r_{32}}\right]</math>

Finally, we get that the electrostatic potential energy stored in the system of three charges:

:<math>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]</math>

|}

== Energy stored in an electrostatic field distribution ==

The energy density, or energy per unit volume, <math>\frac{dU}{dV}</math>, of the [[electrostatic field]] of a continuous charge distribution is:

: <math> u_e = \frac{dU}{dV} = \frac{1}{2} \varepsilon_0 \left|{\mathbf{E}}\right|^2.</math>

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Outline of proof
|-
|
One may take the equation for the electrostatic [[potential energy]] of a continuous charge distribution and put it in terms of the [[electrostatic field]].

Since [[Gauss's law]] for electrostatic field in differential form states

:<math>\mathbf{\nabla}\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}</math>

where

* <math>\mathbf{E} \ </math> is the electric field vector
* <math>\rho \ </math> is the total [[charge density]] including [[dipole]] charges [[bound charge|bound]] in a material
* <math>\varepsilon_0 </math> is the [[permittivity of free space]],

then,

: <math>
\begin{align}
U & = \frac{1}{2}\int \limits_{\text{all space}} \rho(r) \Phi(r) \, dV \\
& = \frac{1}{2}\int \limits_{\text{all space}} \varepsilon_0(\mathbf{\nabla}\cdot{\mathbf{E}})\Phi \, dV
\end{align}
</math>

so, now using the following divergence vector identity

:<math> \nabla\cdot(\mathbf{A}{B}) = (\nabla\cdot\mathbf{A}){B} + \mathbf{A}\cdot(\nabla{B}) \Rightarrow (\nabla\cdot\mathbf{A}){B} = \nabla\cdot(\mathbf{A}{B}) - \mathbf{A}\cdot(\nabla{B})</math>

we have

: <math> U = \frac{\varepsilon_0}{2}\int \limits_{\text{all space}} \mathbf{\nabla}\cdot(\mathbf{E}\Phi) dV - \frac{\varepsilon_0}{2}\int \limits_{\text{all space}} (\mathbf{\nabla}\Phi)\cdot\mathbf{E} dV</math>

using the [[divergence theorem]] and taking the area to be at infinity where <math>\Phi(\infty) = 0</math>

: <math>
\begin{align}
U & = \overbrace{\frac{\varepsilon_0}{2}\int\limits_{{}^\text{boundary}_\text{ of space}} \Phi\mathbf{E}\cdot d\mathbf A}^{0} - \frac{\varepsilon_0}{2}\int \limits_{\text{all space}} (-\mathbf{E})\cdot\mathbf{E} \, dV \\
& = \int \limits_{\text{all space}} \frac{1}{2}\varepsilon_0\left|{\mathbf{E}}\right|^2 \, dV.
\end{align}
</math>

So, the energy density, or energy per unit volume <math>\frac{dU}{dV}</math> of the [[electrostatic field]] is:

: <math> u_e = \frac{1}{2} \varepsilon_0 \left|{\mathbf{E}}\right|^2.</math>

|}

== Energy stored in electronic elements==
[[File:Electronic component electrolytic capacitors.jpg|right|thumb|150x150px|The electric potential energy stored in a [[capacitor]] is U<sub>E</sub>=½ CV<sup>2</sup>]]
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's first law|Joule effect]]. A [[capacitor]] stores it in its electric field. The total electric potential energy stored in a capacitor is given by

: <math> U_E = \frac{1}{2}QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C}</math>

where ''C'' is the [[capacitance]], ''V'' is the [[electric potential]] difference, and ''Q'' the [[Electric charge|charge]] stored in the capacitor.

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Outline of proof
|-
|
One may assemble charges to a capacitor in infinitesimal increments, <math>dq \to 0</math>, such that the amount of work done to assemble each increment to its final location may be expressed as

: <math> W_q = V dq = \frac{q}{C}dq.</math>

The total work done to fully charge the capacitor in this way is then

: <math> W = \int dW = \int_0^Q V dq = \frac{1}{C} \int_0^Q qdq = \frac{Q^2}{2C}.</math>

where <math>Q</math> is the total charge on the capacitor. This work is stored as electrostatic potential energy, hence,

: <math> W = U_E = \frac{Q^2}{2C}.</math>

Notably, this expression is only valid if <math>dq \to 0</math>, which holds for many-charge systems such as large capacitors having metallic electrodes. For few-charge systems the discrete nature of charge is important. The total energy stored in a few-charge capacitor is

: <math> U_E = \frac{Q^2}{C}</math>

which is obtained by a method of charge assembly utilizing the smallest physical charge increment <math>\Delta q = e</math> where <math>e</math> is the [[Elementary charge|elementary unit of charge]] and <math>Q = Ne</math> where <math>N</math> is the total number of charges in the capacitor.

|}

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

: <math>U_E = \frac{1}{2}\int_V \mathrm{E}\cdot \mathrm{D} dV</math>

where <math>\mathrm{D}</math> 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, <math>\rho</math>,

: <math>U_E = \frac{1}{2} \int_V \rho \Phi dV</math>

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 (<math>dq \to 0</math>) such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.

==Notes==
{{reflist|group=note}}

== References ==
{{reflist}}

{{Footer energy}}

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