Electrostatics

by electromagnetics.info


Often electromagnetic fields tend to a steady state. Consider for example a circuit consisting of a cell and an ideal capacitor and a switch joined by conducting wire. When the switch is turned to on the current flows on to the capacitor plates. However, as the capacitor charges, the current reduces. When the capacitor is fully charged, no current flows. A steady or electrostatic state has been reached.

When an electromagnetic field has reached a steady state, the electromagnetic properties are time-invariant. To simplify matters let us work from Maxwell's equations for linear isotropic media. The properties of the field can be determined from the third of Maxwell's equations and putting all derivatives with respect to time equal to zero and there is no current (s = 0):

Ñ×E = 0
(1)

Ñ×H = 0
(2)

Ñ. H = 0
(3)

Ñ. E = r
e
(4)

Equation (1) tells us that E is an irrotational vector and is expressible in terms of the gradient of a scalar f

E = Ñf  .
(5)
Substituting this into equation (4) gives Poisson's Equation

Ñ2 f = r
e
   .
(6)
In regions where there is no charge (or when charge lies on the boundaries of a region) then equation (6) becomes Laplace's Equation

Ñ2 f = 0  .
(7)

If charge lies on the bounding surfaces of the domain then from equation (4)

E.n = r
e
 ,
where n is the unit normal to the surface, pointing into the domain. Since E = Ñf then (Ñf).n = r/e or f/ n = r/e.

Poisson's and Laplace's equations are elliptic partial differential equations. For general electrostatic domains, Poisson's and Laplace's equations can be solved using the finite difference method (FDM) using a grid over the domain or finite element method (FEM) with a mesh of the domain. The boundary element method (BEM) can also be used to solve Laplace's equation directly using a mesh over the surface of the boundaries within the domain. Poisson's equation can also be solved by the BEM but the method is more compicated.