Electric potential energy

Short description: Potential energy that results from conservative Coulomb forces

Electric potential energy

Common symbols

| UE
SI unit| joule (J)

Derivations from
other quantities

| UE = C · V2 / 2
Part of a series of articles about

Electromagnetism

* Electricity
* Magnetism

Electrostatics

* Conductor
* Coulomb's law
* Electric charge
* Electric dipole moment
* Electric field
* Electric flux / potential energy
* Electrostatic discharge
* Gauss's law
* Induction
* Insulator
* Polarization density
* Static electricity
* Triboelectricity

Magnetostatics

* Ampère's law
* Biot–Savart law
* Gauss's law for magnetism
* Magnetic field
* Magnetic flux
* Magnetic dipole moment
* Magnetization
* Magnetomotive force

Electrodynamics

* Lorentz force law
* Electromagnetic induction
* Faraday's law
* Lenz's law
* Displacement current
* Magnetic potential
* Maxwell's equations
* Electromagnetic field
* Electromagnetic pulse
* Electromagnetic radiation
* Maxwell tensor
* Poynting vector
* Liénard–Wiechert potential
* Jefimenko's equations
* Eddy current
* London equations

* Mathematical descriptions of the electromagnetic field

Electrical network

* Alternating current
* Capacitance
* Direct current
* Electric current
* Electric potential
* Electromotive force
* Impedance
* Inductance
* Ohm's law
* Parallel circuit
* Resistance
* Resonant cavities
* Series circuit
* Voltage
* Waveguides

Covariant formulation

* Electromagnetic tensor
(stress–energy tensor)

* Four-current
* Electromagnetic four-potential

Scientists

* Ampère
* Biot
* Coulomb
* Faraday
* Gauss
* Heaviside
* Henry
* Hertz
* Lenz
* Lorentz
* Maxwell
* Ørsted
* Savart
* Tesla
* Volta
* Weber

* v
* t
* e

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 be said to have electric potential energy by virtue of either its own electric charge or 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.

## Contents

* 1 Definition
* 2 Units
* 3 Electrostatic potential energy of one point charge
* 3.1 One point charge q in the presence of another point charge Q
* 3.2 One point charge q in the presence of n point charges Qi
* 4 Electrostatic potential energy stored in a system of point charges
* 4.1 Energy stored in a system of one point charge
* 4.2 Energy stored in a system of two point charges
* 4.3 Energy stored in a system of three point charges
* 5 Energy stored in an electrostatic field distribution in vacuum
* 6 Energy stored in electronic elements
* 7 Notes
* 8 References
* 9 External links

## 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, 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

UE(𝐫)=−Wrref->r=−∫𝐫ref𝐫q𝐄(𝐫′)⋅d𝐫′

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 Φ is defined as the product of the charge and the electric potential.

UE(𝐫)=qΦ(𝐫)

where Φ is the electric potential generated by the charges, which is a function of position r.

## 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

A point charge q in the electric field of another 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:

UE(r)=keqQr,

where ke=14πε0 is the Coulomb 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.

Outline of proof

The electrostatic force F acting on a charge q can be written in terms of the electric field E as 𝐅=q𝐄,

By definition, the change in electrostatic potential energy, UE, of a point charge q that has moved from the reference position rref 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 rref to that position r.

UE(r)−UE(rref)=−Wrref->r=−∫rrefrq𝐄⋅d𝐬.

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 rref to r,
* Wrref->r is the work done by the electrostatic force to bring the charge from the reference position rref to r,

Usually UE is set to zero when rref is infinity: UE(rref=∞)=0 so UE(r)=−∫∞rq𝐄⋅d𝐬

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 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 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:

𝐄⋅d𝐬=|𝐄|⋅|d𝐬|cos⁡(0)=Eds

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

|𝐄|=E=14πε0Qs2

and the integral can be easily evaluated:

UE(r)=−∫∞rq𝐄⋅d𝐬=−∫∞r14πε0qQs2ds=14πε0qQr=keqQr

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

Electrostatic potential energy of q due to Q1 and Q2 charge system:UE=q14πε0(Q1r1+Q2r2)

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:

UE(r)=keq∑i=1nQiri,

where ke=14πε0 is the Coulomb constant, ri is the distance between the point charges q and Qi, and q and Qi are the assigned 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:

UE=12∑i=1NqiΦ(𝐫i)=12ke∑i=1Nqi∑j≠ij=1Nqjrij,

|
|

(1)

where, for each i value, Φ(ri) is the electrostatic potential due to all point charges except the one at ri,[note 2] and is equal to: Φ(𝐫i)=ke∑j≠ij=1Nqj𝐫ij, where rij is the distance between qi and qj.

Outline of proof

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 UE=q2Φ1(𝐫2).

Doing the same calculation with respect to the other charge, we obtain UE=q1Φ2(𝐫1).

The electrostatic potential energy is mutually shared by q1 and q2, so the total stored energy is UE=12[q2Φ1(𝐫2)+q1Φ2(𝐫1)]

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:

UE=12∑i=1NqiΦ(𝐫i).

### 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, Q1. The electric potential Φ(r) due to Q1 is Φ(r)=keQ1r

Hence we obtain, the electrostatic potential energy of q in the potential of Q1 as UE=14πε0qQ1r1 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: UE=14πε0[Q1Q2r12+Q1Q3r13+Q2Q3r23]

Outline of proof

Using the formula given in (1), the electrostatic potential energy of the system of the three charges will then be: UE=12[Q1Φ(𝐫1)+Q2Φ(𝐫2)+Q3Φ(𝐫3)]

Where Φ(𝐫1) is the electric potential in r1 created by charges Q2 and Q3, Φ(𝐫2) is the electric potential in r2 created by charges Q1 and Q3, and Φ(𝐫3) is the electric potential in r3 created by charges Q1 and Q2. The potentials are:

Φ(𝐫1)=Φ2(𝐫1)+Φ3(𝐫1)=14πε0Q2r12+14πε0Q3r13 Φ(𝐫2)=Φ1(𝐫2)+Φ3(𝐫2)=14πε0Q1r21+14πε0Q3r23 Φ(𝐫3)=Φ1(𝐫3)+Φ2(𝐫3)=14πε0Q1r31+14πε0Q2r32

Where rij is the distance between charge Qi and Qj.

If we add everything:

UE=1214πε0[Q1Q2r12+Q1Q3r13+Q2Q1r21+Q2Q3r23+Q3Q1r31+Q3Q2r32]

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

UE=14πε0[Q1Q2r12+Q1Q3r13+Q2Q3r23]

## Energy stored in an electrostatic field distribution in vacuum

The energy density, or energy per unit volume, dUdV, of the electrostatic field of a continuous charge distribution is: ue=dUdV=12ε0|𝐄|2.

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 ∇⋅𝐄=ρε0 where

* 𝐄 is the electric field vector
* ρ is the total charge density including dipole charges bound in a material
* ε0 is the permittivity of free space,

then, U=12∫all spaceρ(r)Φ(r)dV=12∫all spaceε0(∇⋅𝐄)ΦdV

so, now using the following divergence vector identity

∇⋅(𝐀B)=(∇⋅𝐀)B+𝐀⋅(∇B)⇒(∇⋅𝐀)B=∇⋅(𝐀B)−𝐀⋅(∇B)

we have

U=ε02∫all space∇⋅(𝐄Φ)dV−ε02∫all space(∇Φ)⋅𝐄dV

using the divergence theorem and taking the area to be at infinity where Φ(∞)=0

U=ε02∫ of spaceboundaryΦ𝐄⋅d𝐀⏞0−ε02∫all space(−𝐄)⋅𝐄dV=∫all space12ε0|𝐄|2dV.

So, the energy density, or energy per unit volume dUdV of the electrostatic field is:

ue=12ε0|𝐄|2.

## Energy stored in electronic elements

The electric potential energy stored in a capacitor is UE=⁠1/2⁠ CV2

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 UE=12QV=12CV2=Q22C where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.

Outline of proof

One may assemble charges to a capacitor in infinitesimal increments, dq->0, such that the amount of work done to assemble each increment to its final location may be expressed as

Wq=Vdq=qCdq.

The total work done to fully charge the capacitor in this way is then W=∫dW=∫0QVdq=1C∫0Qqdq=Q22C. where Q is the total charge on the capacitor. This work is stored as electrostatic potential energy, hence, W=UE=Q22C.

Notably, this expression is only valid if dq->0, 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 UE=Q22C which is obtained by a method of charge assembly utilizing the smallest physical charge increment Δq=e where e is the elementary unit of charge and Q=Ne where N 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 UE=12∫VE⋅DdV

where D is the electric displacement field within a dielectric material and integration is over the entire volume of the dielectric.

(A virtual experiment based on the energy transfert between capacitor plates reveals that an additional term must be taken into account when the electrostatic energy is expressed in terms of the electric field and displacement vectors [3].

While this extra energy cancels when dealing with insulators, in general it cannot be ignored, as for instance with semiconductors.)

The total electrostatic potential energy stored within a charged dielectric may also be expressed in terms of a continuous volume charge, ρ, UE=12∫VρΦdV 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 (dq->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. ↑ 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. https://archive.org/details/fundamentalsofp000davi.
3. ↑ Sallese (2016-06-01). "A new constituent of electrostatic energy in semiconductors" (in en). The European Physical Journal B 89 (6): 136. doi:10.1140/epjb/e2016-60865-4. ISSN 1434-6036. https://doi.org/10.1140/epjb/e2016-60865-4.

## External links

* v
* t
* e

Energy

* Outline
* History
* Index

Fundamental concepts|

* Energy
* Units
* Conservation of energy
* Energetics
* Energy transformation
* Energy condition
* Energy transition
* Energy level
* Energy system
* Mass
* Negative mass
* Mass–energy equivalence
* Power
* Thermodynamics
* Quantum thermodynamics
* Laws of thermodynamics
* Thermodynamic system
* Thermodynamic state
* Thermodynamic potential
* Thermodynamic free energy
* Irreversible process
* Thermal reservoir
* Heat transfer
* Heat capacity
* Volume (thermodynamics)
* Thermodynamic equilibrium
* Thermal equilibrium
* Thermodynamic temperature
* Isolated system
* Entropy
* Free entropy
* Entropic force
* Negentropy
* Work
* Exergy
* Enthalpy

Types|

* Kinetic
* Internal
* Thermal
* Potential
* Gravitational
* Elastic
* Electric potential energy
* Mechanical
* Interatomic potential
* Electrical
* Magnetic
* Ionization
* Radiant
* Binding
* Nuclear binding energy
* Gravitational binding energy
* Chromodynamic
* Dark
* Quintessence
* Phantom
* Negative
* Chemical
* Rest
* Sound energy
* Surface energy
* Mechanical wave
* Sound wave
* Vacuum energy
* Zero-point energy

Energy carriers|

* Radiation
* Enthalpy
* Fuel
* fossil fuel
* Heat
* Latent heat
* Work
* Electricity
* Battery
* Capacitor

Primary energy|

* Fossil fuel
* Coal
* Petroleum
* Natural gas
* Nuclear fuel
* Natural uranium
* Radiant energy
* Solar
* Wind
* Hydropower
* Marine energy
* Geothermal
* Bioenergy
* Gravitational energy

Energy system
components|

* Energy engineering
* Oil refinery
* Electric power
* Fossil fuel power station
* Cogeneration
* Integrated gasification combined cycle
* Nuclear power
* Nuclear power plant
* Radioisotope thermoelectric generator
* Solar power
* Photovoltaic system
* Concentrated solar power
* Solar thermal energy
* Solar power tower
* Solar furnace
* Wind power
* Wind farm
* Airborne wind energy
* Hydropower
* Hydroelectricity
* Wave farm
* Tidal power
* Geothermal power
* Biomass

Use and
supply|

* Energy consumption
* Energy storage
* World energy consumption
* Energy security
* Energy conservation
* Efficient energy use
* Transport
* Agriculture
* Renewable energy
* Sustainable energy
* Energy policy
* Energy development
* Worldwide energy supply
* South America
* USA
* Mexico
* Canada
* Europe
* Asia
* Africa
* Australia

Misc.|

* Jevons paradox
* Carbon footprint

* Category
* Commons
* Portal
* WikiProject

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Electric potential energy. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Electric_potential_energy&oldid=372618"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template