Physics:Electric potential energy
Electric potential energy  

Common symbols  U_{E} 
SI unit  joule (J) 
Derivations from other quantities  U_{E} = C · V^{2} / 2 
Part of a series of articles about 
Electromagnetism 

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.
Definition
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.
The electrostatic potential energy can also be defined from the electric potential as follows:
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} Joules. Also electronvolts may be used, 1 eV = 1.602×10^{−19} Joules.
Electrostatic potential energy of one point charge
One point charge q in the presence of another point charge Q
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:
[math]\displaystyle{ U_E(r) = k_e\frac{qQ}{r}, }[/math]
where [math]\displaystyle{ k_e = \frac{1}{4\pi\varepsilon_0} }[/math] 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 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 electrostatic force F acting on a charge q can be written in terms of the electric field E as [math]\displaystyle{ \mathbf{F} = q\mathbf{E} , }[/math]
By definition, the change in electrostatic potential energy, U_{E}, of a point charge q that has moved from the reference position r_{ref} 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_{ref} to that position r.
[math]\displaystyle{ 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 of the position vector,
 ds = differential displacement vector along a path C going from r_{ref} to r,
 [math]\displaystyle{ W_{r_{\rm ref} \rightarrow r } }[/math] is the work done by the electrostatic force to bring the charge from the reference position r_{ref} to r,
Usually U_{E} is set to zero when r_{ref} is infinity: [math]\displaystyle{ U_E (r_{\rm ref}=\infty) = 0 }[/math] so [math]\displaystyle{ U_E(r) =  \int_\infty^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s} }[/math]
When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. This happens in timeinvariant electric fields. When talking about electrostatic potential energy, timeinvariant electric fields are always assumed so, in this case, the electric field is 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 ds must be parallel:
[math]\displaystyle{ \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]\displaystyle{ \mathbf{E} = E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{s^2} }[/math]
and the integral can be easily evaluated:
[math]\displaystyle{ 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_{i}
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:
[math]\displaystyle{ U_E(r) = k_e q \sum_{i=1}^n \frac{Q_i}{r_i}, }[/math]
where [math]\displaystyle{ k_e = \frac{1}{4\pi\varepsilon_0} }[/math] 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.
Electrostatic potential energy stored in a system of point 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:
[math]\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_\stackrel{j=1}{j \ne i}^N \frac{q_j}{r_{ij}}, }[/math] 

( ) 
where, for each i value, Φ(r_{i}) is the electrostatic potential due to all point charges except the one at r_{i},^{[note 2]} and is equal to: [math]\displaystyle{ \Phi(\mathbf{r}_i) = k_e\sum_\stackrel{j=1}{j \ne i}^N \frac{q_j}{\mathbf{r}_{ij}}, }[/math] where r_{ij} is the distance between q_{i} and q_{j}.
The electrostatic potential energy U_{E} 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_{1} generates an electrostatic potential Φ_{1}, which is a function of position r, then [math]\displaystyle{ U_\mathrm{E} = q_2 \Phi_1(\mathbf r_2). }[/math]
Doing the same calculation with respect to the other charge, we obtain [math]\displaystyle{ U_\mathrm{E} = q_1 \Phi_2(\mathbf r_1). }[/math]
The electrostatic potential energy is mutually shared by [math]\displaystyle{ q_1 }[/math] and [math]\displaystyle{ q_2 }[/math], so the total stored energy is [math]\displaystyle{ 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_{E} stored in a system of N charges q_{1}, q_{2}, …, q_{N} at positions r_{1}, r_{2}, …, r_{N} respectively, is:
[math]\displaystyle{ 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 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, Q_{1}. The electric potential Φ(r) due to Q_{1} is [math]\displaystyle{ \Phi(r) = k_e \frac{Q_1}{r} }[/math]
Hence we obtain, the electrostatic potential energy of q in the potential of Q_{1} as [math]\displaystyle{ U_E = \frac{1}{4\pi\varepsilon_0} \frac{q Q_1}{r_1} }[/math] where r_{1} 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_{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: [math]\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] }[/math]
Using the formula given in (1), the electrostatic potential energy of the system of the three charges will then be: [math]\displaystyle{ 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]\displaystyle{ \Phi(\mathbf{r}_1) }[/math] is the electric potential in r_{1} created by charges Q_{2} and Q_{3}, [math]\displaystyle{ \Phi(\mathbf{r}_2) }[/math] is the electric potential in r_{2} created by charges Q_{1} and Q_{3}, and [math]\displaystyle{ \Phi(\mathbf{r}_3) }[/math] is the electric potential in r_{3} created by charges Q_{1} and Q_{2}. The potentials are:
[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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_{ij} is the distance between charge Q_{i} and Q_{j}.
If we add everything:
[math]\displaystyle{ 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]\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] }[/math]
Energy stored in an electrostatic field distribution in vacuum
The energy density, or energy per unit volume, [math]\displaystyle{ \frac{dU}{dV} }[/math], of the electrostatic field of a continuous charge distribution is: [math]\displaystyle{ u_e = \frac{dU}{dV} = \frac{1}{2} \varepsilon_0 \left{\mathbf{E}}\right^2. }[/math]
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]\displaystyle{ \mathbf{\nabla}\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0} }[/math] where
 [math]\displaystyle{ \mathbf{E} }[/math] is the electric field vector
 [math]\displaystyle{ \rho }[/math] is the total charge density including dipole charges bound in a material
 [math]\displaystyle{ \varepsilon_0 }[/math] is the permittivity of free space,
then, [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \Phi(\infty) = 0 }[/math]
[math]\displaystyle{ \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]\displaystyle{ \frac{dU}{dV} }[/math] of the electrostatic field is:
[math]\displaystyle{ u_e = \frac{1}{2} \varepsilon_0 \left{\mathbf{E}}\right^2. }[/math]
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 electrostatic potential energy stored in a capacitor is given by [math]\displaystyle{ 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 charge stored in the capacitor.
One may assemble charges to a capacitor in infinitesimal increments, [math]\displaystyle{ dq \to 0 }[/math], such that the amount of work done to assemble each increment to its final location may be expressed as
[math]\displaystyle{ W_q = V \, dq = \frac{q}{C}dq. }[/math]
The total work done to fully charge the capacitor in this way is then [math]\displaystyle{ W = \int dW = \int_0^Q V \, dq = \frac{1}{C} \int_0^Q q \, dq = \frac{Q^2}{2C}. }[/math] where [math]\displaystyle{ Q }[/math] is the total charge on the capacitor. This work is stored as electrostatic potential energy, hence, [math]\displaystyle{ W = U_E = \frac{Q^2}{2C}. }[/math]
Notably, this expression is only valid if [math]\displaystyle{ dq \to 0 }[/math], which holds for manycharge systems such as large capacitors having metallic electrodes. For fewcharge systems the discrete nature of charge is important. The total energy stored in a fewcharge capacitor is [math]\displaystyle{ U_E = \frac{Q^2}{C} }[/math] which is obtained by a method of charge assembly utilizing the smallest physical charge increment [math]\displaystyle{ \Delta q = e }[/math] where [math]\displaystyle{ e }[/math] is the elementary unit of charge and [math]\displaystyle{ Q = Ne }[/math] where [math]\displaystyle{ 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]\displaystyle{ U_E = \frac{1}{2} \int_V \mathrm{E} \cdot \mathrm{D} \, dV }[/math]
where [math]\displaystyle{ \mathrm{D} }[/math] 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, [math]\displaystyle{ \rho }[/math], [math]\displaystyle{ 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]\displaystyle{ dq \to 0 }[/math]) such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.
Notes
References
 ↑ Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN:0471927120
 ↑ Halliday, David; Resnick, Robert; Walker, Jearl (1997). "Electric Potential". Fundamentals of Physics (5th ed.). John Wiley & Sons. ISBN 0471105597. https://archive.org/details/fundamentalsofp000davi.
External links
Original source: https://en.wikipedia.org/wiki/Electric potential energy.
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