NationMaster - Encyclopedia: Electric potential energy

The electric potential energy $U E$ is the potential energy associated with the conservative Coulomb forces between charged particles contained within a system, where the reference potential energy is usually chosen to be zero for particles at infinite separation.^[1]^:§25-1 It can be defined as the amount of work one must apply to (massless) charged particles to bring them from infinite separation to some finite proximity configuration. This is also equal to the negative of the work of the Coulomb forces that the particles exert on each other during the quasistatic move:^[1]^:§25-1 Potential energy can be thought of as energy stored within a physical system. ... A conservative force is a force which is path-independent. ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrical force that one stationary, electrically charged substance of small volume (ideally, a point source) exerts on another. ... In physics, a charged particle is a particle with an electric charge. ... A physical system is a system that is comprised of matter and energy. ... Look up work in Wiktionary, the free dictionary. ... In thermodynamics a quasistatic process is a process that happens infinitely slowly. ...

$U_E = W_{app} = - W_infty$ .

W a p p

is the work required to bring the system to a certain finite proximity configuration. "app" stands for applied, because this is work that must be applied to the system (or be supplied by another form of energy contained by the system) to configure it

is the work done by electrostatic inter-particle Coulomb forces during the move from infinity.

1 Disambiguation
2 Units
3 Properties
4 Potential energy stored in a configuration of discrete charges
5 Potential energy of a uniform charge distribution
6 Energy stored in an electric field
7 See also
8 References

Electrostatics is the branch of physics that deals with the force exerted by a static (i. ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrical force that one stationary, electrically charged substance of small volume (ideally, a point source) exerts on another. ...

Disambiguation

Sometimes people refer to the potential energy of a charge in an electric field. This actually refers to the potential energy of the system containing the charge and the other particles that created the electric field.^[1]^:§25-1 In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ...

Furthermore, to calculate the work required to bring a charged particle into the vicinity of other particles, it is sufficient to know only the field generated by the other particles and the charge of the particle being moved. The field of the moving particle and the individual charges of the other particles do not need to be known.

Finally, it must be stressed that, even though this article talks about moving particles, the Coulomb force law on which this discussion is based only holds in the case of electrostatic systems. Therefore, any movement would have to be a quasistatic process. In thermodynamics a quasistatic process is a process that happens infinitely slowly. ...

Units

The SI unit of electrical energy is the joule. In the context of use of electrical energy for lighting, heating, motors, and other applications, larger units such as the kilowatt-hour, equivalent to 3.6 million joules, are used. â€œSIâ€ redirects here. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... The watt-hour (symbol WÂ·h) is a unit of energy. ...

Properties

Since Coulomb forces are conservative, change in electric potential energy is independent of the path the system takes between two configurations. Correspondingly, the work done by the forces is also path independent and gives the change in electric potential energy:^[1]^:§25-1

U f - U i = W a p p = - W c

U f

is the electric potential energy at the final state

U i

is the electric potential energy at the initial state

W a p p

is the work required to bring the system to the final state from the initial state

W c

is the work done by inter-particle Coulomb forces during the move

The usual choice of datum is $U i = 0$ at infinite particle separation, as stated in the opening.

Due to the proportionality of the Coulomb force with the charges of the particles on which it acts, the potential energy of a charged particle in a given electric field is also proportional to the charge of the particle. So, the potential energy may be normalized by the charge of the particle on which the field acts to give a quantity called the electric potential, which is only a function of the strength of the electric field (and the particle location):^[1]^:§25-1 This article does not cite any references or sources. ...

$V = frac {U_E}{q} = frac {W_{app}}{q} = - frac {W_infty}{q}$

V

is the electric potential of the field

q

is the charge of the particle in the field

The above equation is the definition of electric potential for a system (with a datum of zero potential energy at infinite particle separation). It is very important to note that the electric field under consideration is not the total field, but the field due only to all particles except the one at the location for which the electric potential energy (and thus electric potential) is calculated.

Potential energy stored in a configuration of discrete charges

The potential energy between two charges is equal to the potential energy of one charge in the electric field of the other. That is to say, if $q 1$ generates a scalar electric potential field , which is a function of position $mathbf r$ , then $U_E = q_2 V_1(mathbf r_2)$ . Also, a similar development gives .

This can be generalized to give an expression for a group of N charges, $q i$ at positions $mathbf r_i$ :

$U_E = frac{1}{2}sum_i^N q_i V(mathbf r_i)$

$V(mathbf r_i)$ refers to the electric field due to all particles except the one at $mathbf r_i$

Note: The factor of one half accounts for the 'double counting' of charge pairs. For example, consider the case of just two charges.

Alternatively, the factor of one half may be dropped if the sum is only performed once per particle pair. This is done in the examples below to cut down on the math.

One charged particle

The electric potential energy of a system containing only one point charge is zero, as no energy is required to move the charge particle from infinity to its location.

Two charged particles

Consider bringing a second charge into position:

where

k_e is Coulomb's constant

q₁, q₂ are the two charges

r is the distance between the two particles

The electric potential energy will be negative if the charges have opposite sign and positive if the charges have the same sign. This simply means that potential energy is lost by a system of opposite charges moving together, which can be explained as 'opposite charges attract'. It has been suggested that this article or section be merged into Electrostatic force. ...

Three or more charged particles

For 3 or more point charges, the electric potential energy of the system may be calculated by bringing individual charges into position 1 after another, and taking the sum of energy required to bring the additional charge into position.

$U_mathrm{E} = k_e left({frac{q_1 q_2}{r_{12}}} + {frac{q_1 q_3}{r_{13}}} + {frac{q_2 q_3}{r_{23}}} + ...right)$

where

k_e is Coulomb's constant

q₁, q₂, ..., are the charges

r_mn is the distance between two particles, m and n (e.g. r₁₂).

Potential energy of a uniform charge distribution

The previous equation can again be generalized to give an expression of the potential energy of a uniform charge distribution.

$U = frac{1}{2}int limits_{text{all space}} rho(r)V(r)d^3r$

where:

ρ(r)

is the charge density of the distribution.

V (r)

is the electric potential at position r.

Charge density is the amount of electric charge per unit volume. ... This article does not cite any references or sources. ...

Energy stored in an electric field

One may take the equation for the potential energy of a uniform charge distribution and put it in terms of the electric field. Potential energy can be thought of as energy stored within a physical system. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ...

Since

$mathbf{nabla}cdotmathbf{E} = frac{rho}{epsilon_o}$

where

ε o

is the permittivity of the medium

E is the electric field vector.

then, Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

$U = frac{1}{2}int limits_{text{all space}} rho(r)V(r)d^3r$

$= frac{1}{2}int limits_{text{all space}} epsilon_o(mathbf{nabla}cdot{mathbf{E}})V(r)d^3r$

so, now using the following divergence vector identity

we have

using the divergence theorem and taking the area to be at infinity where $V(infty) = 0$ In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ...

$U = frac{epsilon_o}{2}int Vmathbf{E}cdot dA - frac{epsilon_o}{2}int limits_{text{all space}} (-mathbf{E})cdotmathbf{E} d^3r$

$= int limits_{text{all space}} frac{1}{2}epsilon_oleft|{mathbf{E}}right|^2 d^3r$

So, the energy density, or energy per unit volume of the electric field is: In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ...

$u_e = frac{1}{2} epsilon_o left|Eright|^2$

References

^ ^a ^b ^c ^d ^e Halliday, David; Resnick, Robert; Walker, Jearl (1997). Fundamentals of Physics, 5th (in English), John Wiley & Sons. ISBN 0-471-10559-7.

Categories: Energy | Electrostatics

Results from FactBites:

Electric potential - definition of Electric potential in Encyclopedia (774 words)
	Electrical potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential or the electric potential, typically measured in volts.
	The electrical potential is therefore measured in units of energy per unit of electric charge.
	As the object falls, that potential energy decreases and is translated to motion, or inertial energy.

Electric potential - Wikipedia, the free encyclopedia (1120 words)
	Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts.
	The electric potential is therefore measured in units of energy per unit of electric charge.
	The electric potential and the magnetic vector potential together form a four vector, so that the two kinds of potential is mixed under Lorentz transformations.

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Contents

Disambiguation GA_googleFillSlot("encyclopedia_square");

Units

Properties

Potential energy stored in a configuration of discrete charges

One charged particle

Two charged particles

Three or more charged particles

Potential energy of a uniform charge distribution

Energy stored in an electric field

See also

References

Disambiguation