Electric potential energy

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Electric energy 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

$U_E = W_{app} = - W_\infty$

where

$W_{app} \$ 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
$W_\infty \$ is the work done by electrostatic inter-particle Coulomb forces during the move from infinity.

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

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.

[edit] 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 $V_1(\mathbf r)$ , which is a function of position $\mathbf r$ , then $U_E = q_2 V_1(\mathbf r_2)$ . Also, a similar development gives $U_E = q_1 V_2(\mathbf r_1)$ .

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

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

where $V(\mathbf r_i)$ 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.

[edit] 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.

[edit] Two charged particles

Consider bringing a second charge into position:

$U_\mathrm{E} = k_e \frac{q_1 q_2}{r}$

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'.

[edit] 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 one 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).

[edit] Potential energy of a continuous charge distribution

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

[edit] Energy stored in an electric field

One may take the equation for the potential energy of a continuous charge distribution and put it in terms of the electric field.

Since Gauss' law for electric field in differential form states

$\mathbf{\nabla}\cdot\mathbf{E} = \frac{\rho}{\epsilon_0}$

where

$\mathbf{E} \$ is the electric field vector
$\rho \$ is the total charge density including dipole charges bound in a material
$\epsilon_0 \$ is the vacuum permittivity.

then,

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

$= \frac{1}{2}\int \limits_{\text{all space}} \epsilon_0(\mathbf{\nabla}\cdot{\mathbf{E}})V(r)d^3r$

so, now using the following divergence vector identity

$\nabla\cdot(\vec{A}{B}) = (\nabla\cdot\vec{A}){B} + \vec{A}\cdot(\nabla{B}) \Rightarrow (\nabla\cdot\vec{A}){B} = \nabla\cdot(\vec{A}{B}) - \vec{A}\cdot(\nabla{B})$

we have

$U = \frac{\epsilon_0}{2}\int \limits_{\text{all space}} \mathbf{\nabla}\cdot(\mathbf{E}V) d^3r - \frac{\epsilon_0}{2}\int \limits_{\text{all space}} (\mathbf{\nabla}V)\cdot\mathbf{E} d^3r$

using the divergence theorem and taking the area to be at infinity where $V(\infty) = 0$

$U = \frac{\epsilon_0}{2}\int V\mathbf{E}\cdot dA - \frac{\epsilon_0}{2}\int \limits_{\text{all space}} (-\mathbf{E})\cdot\mathbf{E} d^3r$

$= \int \limits_{\text{all space}} \frac{1}{2}\epsilon_0\left|{\mathbf{E}}\right|^2 d^3r.$

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

$u_e = \frac{1}{2} \epsilon_0 \mathbf{E}^2.$

The last two equations show that the energy stored in an electric field should be positive. But it is not always so with elementary particles, in Quantum Mechanics. As an example let us take a hydrogen atom. Hydrogen atom consists of two bound elementary charged particles, proton and electron. Both of them produce electric field. The energy stored in this field consists of the energy stored in the electric field of each particle and an interaction energy. In Quantum Mechanics the energy stored in the electric field of elementary charge particle is always zero, because there is no self-action in Quantum Mechanics. From this follows that the total energy stored in the electric field of the regarded system is equal to the interaction energy of the two charged particles. As the two particles have charges of the opposite signs, this energy is negative.

[edit] See also

[edit] References

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

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[HRW1997-0] Halliday, David; Resnick, Robert; Walker, Jearl (1997) (in English). Fundamentals of Physics (5th ed.). John Wiley & Sons. ISBN 0-471-10559-7.

[1]

Encyklopedia

Electric potential energy

From Wikipedia, the free encyclopedia

Contents

[edit] Potential energy stored in a configuration of discrete charges

[edit] One charged particle

[edit] Two charged particles

[edit] Three or more charged particles

[edit] Potential energy of a continuous charge distribution

[edit] Energy stored in an electric field

[edit] See also

[edit] References