Electrostatic potential energy

The electrostatic potential energy (or simply electrostatic energy ) of an electric charge $q \,$ placed in a point $P \,$ bathing in an electric Potential $V \,$ is defined like the Travail required to transport this load since the infinite one until the position $P \,$ . It is thus worth

E_p=qV (P)
\,

If one places oneself in the case or the sources generating the electric potential V are distributed in an area Bornée of space what makes it possible to allot a zero value of the potential ad infinitum.

Case of a distribution

Electrical potential energy of a distribution $\ rho (P) \,$ of electric charges is then defined as necessary work to transport the whole of the loads which compose it since the infinite one until their final position. Summoning all the contributions one finds then

E_p= {1 \ over2} \ iiint \ frac {\ rho (x_1) \ rho (x_2)}{4 \ pi \ epsilon_0 \|x_2-x_1 \|} {\ rm D} ^3x_1 {\ rm D} ^3x_2
\,

where $\ epsilon_0 \,$ is the dielectric Constante vacuum and the summations being carried out on the extent $\ mathcal {D} \,$ of the burden-sharing.

One can show that one can also obtain the latter by considering the Electric field $\ mathcal {E} \,$ created by the whole of these loads

E_p= {\ epsilon_0 \ over2} \ iiint \ mathcal {E}. \ mathcal {E} {\ rm D} ^3x
\,

integration being done this time on all space.

Several distributions, energy of interaction

In the case of two distributions of limited extent $\ mathcal {D} _1 \,$ and $\ mathcal {D} _2 \,$ characterized by charge distributions $\ rho_1 \,$ and $\ rho_2 \,$ then one can distinguish 3 contributions in total electrical energy

Total E_ {} =E_p^1+E_p^2+E_ {int}
\,

where $E_p^i \,$ obtained consequently formula written previously is famous energies of Coil-interaction and $E_ {int} \,$ is called the potential energy of interaction and is given by

E_ {int} = {1 \ over 4 \ pi \ epsilon_0} \ iiint_ {x_1 \ in \ mathcal {D} _1; x_2 \ in \ mathcal {D} _2} \ frac {\ rho_1 (x_1) \ rho_2 (x_2)}{\|x_2-x_1 \|} {\ rm D} ^3x_1 {\ rm D} ^3x_2= \ epsilon_0 \ iiint \ mathcal {E} _1. \ mathcal {E} _2 {\ rm D} ^3x
\,

with $\ mathcal {E} _ {1,2} \,$ individual electric fields created by each distribution.

This formula is modified when one considers a Magnetic field and more generally when one leaves the framework of the electrostatic (see the electromagnetic article energy).

It is necessary to note finally when if the two distributions $\ mathcal {D} _1 \,$ and $\ mathcal {D} _2 \,$ is Localisées in two points $P_1 \,$ and $P_2 \,$ and together with loads $q_1 \,$ and $q_2 \,$ then coil-energies is divergent but the energy of interaction, it, is well defined and one finds precisely

E_ {int} = {1 \ over 4 \ pi \ epsilon_0} {q_1q_2 \ over \|P_1P_2 \|} =q_1V_2 (P_1) =q_2V_1 (P_2)
\,

See too

Electric field
electric Electromagnetism
potential Energy
Potential

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