Electrostatics toolbox

The starting point is the Maxwell equations.

1. Quasi static approximation, Electrostatics

A consequence of Maxwell’s equations is that changes in time of currents and charges are not synchronized with changes of the electromagnetic fields. There is a delay between the changes of the sources and the changes of the fields. The electromagnetic waves propagate at finite speed. If we ignore this effect, we obtain the electromagnetic fields by considering stationary currents at every instant. In other words, it is the study of electromagnetism with charges at rest.

It is called the quasi-static approximation also called electrostatics and magnetostatics approximations

The approximation is valid provided that the variations in time are small or absent and that the studied geometries are considerably smaller than the wavelength. The quasi-static approximation implies that the equation of continuity can be written as

\[\nabla \cdot \mathbf{j} = 0\]

and that the time derivative of the electric displacement \(\partial D/ \partial t\) can be disregarded in Maxwell-Ampère’s law.

2. Notations and Units

Table 1. Name and units of symbols and variables of the Maxwell’s equations
Notation Quantity Unit SI

\(\mathbf{E}(\mathbf{x},t)\)

electric field intensity

\(V\cdot m^{-1}\)

\(kg\cdot m \cdot s^{-3}\cdot A^{-1}\)

\(V(\mathbf{x},t)\)

electric potential

\(V\)

\(kg\cdot m^2 \cdot s^{-3}\cdot A^{-1}\)

\(\mathbf{D}(\mathbf{x},t)\)

electric flux density

\(C\cdot m^{-2}\)

\(A\cdot s\cdot m^{-2}\)

\(\mathbf{J}(\mathbf{x},t)\)

electric current density

\(A\cdot m^{-2}\)

\(A\cdot m^{-2}\)

\(\rho(\mathbf{x},t)\)

electric charge density

\(C\cdot m^{-3}\)

\(A\cdot s\cdot m^{-3}\)

\(\mathbf{E}_i(\mathbf{x},t)\)

impressed electric field

\(V\cdot m^{-1}\)

\(kg\cdot m \cdot s^{-3}\cdot A^{-1}\)

\(\mathbf{J}_i(\mathbf{x},t)\)

impressed electric current

\(A\cdot m^{-2}\)

\(A\cdot m^{-2}\)

\(\sigma(\mathbf{x},t)\)

conductivity

\(S\cdot m^{-1}\)

\(kg^{-1}\cdot m^{-3}\cdot s^3\cdot A^2\)

\(\epsilon_0(\mathbf{x},t)\)

permittivity of vacuum

\(F\cdot m^{-1}\)

\(kg^{-1}\cdot m^{-3}\cdot s^4\cdot A^2\)

3. Equations

Consider that

  • we are in quasi-electrostatics approximation,

  • we have \(N\) conducting materials whose respective domains are denoted \(\Omega_i,i=1...N\) and their electric conductivity \(\sigma_i\).

The electric potential is solution of the following equation

\[- \nabla \cdot \left(\sigma_i V\right) = \frac{q}{\varepsilon_0}, \quad \mbox{ in } \Omega_i, i=1...N\]

where \(q\) is the charge density and \(\varepsilon_0\) is the vacuum permittivity. This relationship is a form of Poisson’s equation. In the absence of electric charge, the equation becomes Laplace’s equation:

\[\nabla \cdot \left( \sigma_i \nabla V \right) =0, \quad \mbox{ in } \Omega_i, i=1,...,N\]

We have the following relations:

\[\mathbf{E} = -\nabla{V},\quad \mathbf{j} = - \sigma\nabla{V}.\]

where \(\sigma=\sigma_i,i=1...N\).

4. Boundary Conditions

The boundary conditions can be of three types: Dirichlet(Essential), Neumann(Natural) or Integral.

4.1. Dirichlet

Consider two surfaces \(\Gamma_\mbox{in}\) and \(\Gamma_\mbox{out}\). We impose a difference of electric potential.

\[V=V_{\Gamma_\mbox{in}}, \quad V=V_{\Gamma_\mbox{out}}\]

4.2. Neumann

Denote \(\Gamma_N\), the surface where the Neumann condition is imposed.

4.2.1. Insulation

In this case, we have no normal flux, hence we have

\[- \sigma \frac{\partial V}{\partial n} = - \sigma \nabla V \cdot \mathbf{n} = \mathbf{j} \cdot \mathbf{n} = 0\]

4.2.2. Normal flux

\[- \sigma \frac{\partial V}{\partial n} = - \sigma \nabla V \cdot \mathbf{n} = \mathbf{j} \cdot \mathbf{n} = j_N\]

where \(j_N\) is the flux density to be imposed.

4.3. Integral

On \(\Gamma_I\), we wish to impose the current \(I\) (not the current density) which sets at the same time the electric potential \(V_I\) which is, in that particular case, constant but unknown. We have then

\[\int_{\Gamma_I} \mathbf{j} \cdot \mathbf{n} = I_N, \quad V_I \mbox{ is constant}.\]
this boundary condition is only available in the HdG formulation.