Electric potential energy

 Electric potential energy Common symbol(s): UE SI unit: joule (J) Derivations from other quantities: UE = C · V2 / 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 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 electrostatic potential energy, UE, of one point charge q 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 some position r.[1][2]:§25-1[note 2]

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

where F is the electrostatic force, ds is the displacement vector and Φ is the electrostatic 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.

Examples

One point charge q in the presence of n point charges Qn

A point charge q in the electric field of another charge Q.

Let's start first with one point charge q in the presence of only one point charge Q. The electrostatic potential energy, UE, of one point charge q 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 the given formula.

The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qn, 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

In a Two point charges system, Electric potential energy UE of q in the potential well created by Q1 is $U_E = \frac{1}{4\pi\varepsilon_0} \frac{q Q_1}{r}$

The electrostatic potential energy UE 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 q1 generates an electrostatic potential Φ1, which is a function of position r, then

$U_\mathrm{E} = q_2 \Phi_1(\mathbf r_2).$

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

$U_\mathrm{E} = q_1 \Phi_2(\mathbf r_1).$

This can be generalized to say that the electrostatic potential energy UE stored in a system of N charges q1, q2, ..., qN at positions r1, r2, ..., rN respectively, is:

 $U_\mathrm{E} = \frac{1}{2}\sum_{i=1}^N q_i \Phi(\mathbf{r}_i)$,

(1)

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

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.

Two point charges

Consider bringing a point charge, q, into its final position in the vicinity of 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

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

where r1is the separation between the two point charges.

Three point charges

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 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:

$U_\mathrm{E} = \frac{1}{4\pi\varepsilon_0} \left( \frac{Q_1 Q_2}{r_{12}} + \frac{Q_1 Q_3}{r_{13}} + \frac{Q_2 Q_3}{r_{23}} \right)$

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

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

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 the 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.

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.