Electric potential energy

Common symbols

U_{E}

SI unit

joule (J)

Derivations from
other quantities

U_{E} = 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.
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.
Contents

Definition 1

Units 2

Electrostatic potential energy of one point charge 3

One point charge q in the presence of one point charge Q 3.1

One point charge q in the presence of n point charges Qi 3.2

Electrostatic potential energy stored in a system of point charges 4

Energy stored in a system of one point charge 4.1

Energy stored in a system of two point charges 4.2

Energy stored in a system of three point charges 4.3

Energy stored in an electrostatic field distribution 5

Energy in electronic elements 6

Notes 7

References 8
Definition
We define the electric potential energy of a system of point charges 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_{E}, 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 r_{ref}^{[note 1]} to that position r.^{[1]}^{[2]}^{:§251}^{[note 2]}

U_\mathrm{E}(\mathbf r) = W_{r_{\rm ref} \rightarrow r } = \int_^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s},


where E is the electrostatic field and ds is the displacement vector in a curve from the reference position r_{ref} to the final position r.
The electrostatic potential energy can also be defined from the electric potential as follows:

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

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


where \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.
Electrostatic potential energy of one point charge
One point charge q in the presence of one point charge Q
A point charge q in the electric field of another 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:

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 signed values of the charges (not the modules of the charges. For example, 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.

One point charge q in the presence of n point charges Q_{i}
Electrostatic potential energy of q due to Q_{1} and Q_{2} 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, 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:

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, r_{i} is the distance between the point charges q & Q_{i}, and q & Q_{i} are the signed 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:

U_\mathrm{E} = \frac{1}{2}\sum_{i=1}^N q_i \Phi(\mathbf{r}_i) = \frac{1}{2} \sum_{i=1}^N q_i \sum_{j=1}^{N(j\ne i)} k_e \frac{q_j}{r_{ij}},



(1)

where, for each i value, Φ(r_{i}) is the electrostatic potential due to all point charges except the one at r_{i},^{[note 3]} and is equal to:
\Phi(\mathbf{r}_i) = \sum_{j=1}^{N(j\ne i)} k_e \frac{q_j}{\mathbf{r}_{ij}},
where r_{ij} is the distance between q_{j} and r_{i}.

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.
Energy stored in a system of two point charges
Consider bringing a point charge, q, into its final position in the vicinity of a point charge, Q_{1}. The electrostatic potential Φ(r) due to Q_{1} is

\Phi(r) = k_e \frac{Q_1}{r}
Hence we obtain, the electric potential energy of q in the potential of Q_{1} as

U_E = \frac{1}{4\pi\varepsilon_0} \frac{q Q_1}{ r_1 }
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:

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, \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.

Energy in electronic elements
The electric potential energy stored in a
capacitor is U
_{E}=½ CV
^{2}
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.
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 the an external force to bring it from the reference position r_{ref} 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 0471927120

^ Halliday, David; Resnick, Robert; Walker, Jearl (1997). "Electric Potential". Fundamentals of Physics (5th ed.). John Wiley & Sons.
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