Electrostatic Basic principles

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Electrostatics[1][2][3] refers to the physical phenomena related to the presence of electric charges in the objects.

Electric charges[4] are the most basic sources of the electromagnetic nature. Repulsive (or attractive) electrical forces[5][6] are obtained by joining two charges of the same (or different) sign (Coulomb's law).

F = \frac{Q_1Q_2}{4\pi\varepsilon_0 r^2} \,   [Newton] \,


  • Q_1 \, and Q_2 \,   [Coulomb]\, are two electric charges in interaction ;
  • r \,   [meter]\, is the distance between the charges;
  • \varepsilon_0 \,   is a electric constant defined as    \varepsilon_0 \ = 8.854\ \times 10^{-12} [Farad \; · m^{-1}] \; or \; [C^2·N^{-1}·m^{-2}] \,

The image shows the force lines (in blue) produced by two charges, that are the same that those which define the electric field (\vec{E} \; [Volt/m] \,, the electric force per unit of charge associated to each electric charge).

\vec{F} = q\vec{E}\,         E = \frac{Q}{4\pi\varepsilon_0 r^2} \,

For closed surfaces, the total electric flux (\Phi_E \,) is is proportional to the total charge enclosed within the surface (Gauss' law). The electric flux[7] over a surface S is given by the surface integral:

\Phi_E = \int_S \vec{E} \cdot d\vec{A}

This expression can be written in terms of the electric displacement field[8] (\vec{D} \; [C/m^2])\,.

\oint_S \vec{D} \cdot\mathrm{d}\vec{A} =  \int_V\rho\cdot\mathrm{d}V

with \rho \, the volume charge density [C/m^3] \,.

The equation becomes in differential form to our first Maxwell's equation:

\vec{\nabla}\cdot\vec{D} = \rho


\vec{D} = \varepsilon\vec{E} = \varepsilon_0\varepsilon_r\vec{E}

with: \varepsilon, \varepsilon_0, \varepsilon_r the permittivity of the medium, of the free space and relative permittivity, respectively.

The electric field is a conservative vector field. That means that is the gradient of a scalar potential, called electrostatic potential (V [Volts]\,) and that the line integral from one point to another is path independent.

\vec{E}=-\vec{\nabla} V

Therefore, the electric field is also irrotational (our second Maxwell's equation).

\vec{\nabla}\times\vec{E} = 0

By using the first Maxwell's equation (\vec{\nabla}\cdot\vec{D} = \rho) and this definition of electrostatic potential (\vec{E}=-\vec{\nabla} V), we can easily can obtain the Poisson's equation:

\vec{\nabla} \varepsilon \vec{\nabla} V + \rho_v = 0


  1. Sci-Tech Encyclopedia: Electrostatics
  2. wikipedia Electrostatics
  3. wikipedia Static electricity
  4. wikipedia Electric charge
  5. Electric force
  6. Electric force animation
  7. Electric flux
  8. Electric displacement field
  9. Electrostatics Summary
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