Basic Field Equations

Erwin Kasper

1.1 MAXWELL’S EQUATIONS

Maxwell’s equations are partial differential equations for vectorial field functions, which all depend on the spatial position r = (x, y, z) and the time t. In the notation that is familiar in vector analysis, they are given by

Ert=−∂Brt/∂t,

Hrt=∂Drt/∂t+jrt,

ivDrt=ρrt,

ivBrt=0

These vector functions are interrelated by material equations, which describe the electromagnetic properties of matter on a phenomenological basis.

The electric properties are characterized by a polarization P(r, t), the spatial density of electric dipoles:

rt=ε0Ert+Prt.

Similarly, the analogous magnetic property is the magnetization M (r, t), the magnetic dipole density, usually defined by

rt=μ0Hrt+μ0Mrt.

With respect to applications in charged particle optics the polarization P is of little importance, as electrodes are usually conductors, and consequently all static electric fields vanish in these. In the few cases in which insulator materials are present, the assumption of proportionality is sufficient:

rt=εr−ε0Ert,

whereupon Eq. (1.5) reduces to

rt=εrErt.

Even in configurations with a constant dielectric coefficient ε, this becomes a spatial function because it alters discontinuously at any surface. It is now possible to eliminate the vector field D(r, t) from Maxwell’s equations, and we shall do this here.

The analogous linearizations with respect to magnetic fields would result in

rt=μr/μ0−1Hrt,

rt=μrHrt.

It would be a considerable simplification if these relations were valid throughout. We then speak of linear or unsaturated media.

Unfortunately, this assumption does not always hold, and there are then different steps of generalization. The simplest one is the case of isotropic nonlinear media, most favorably presented by

r=vr,|B|Br.

This equation means that H and B always have the same direction, but the material factor v := μ− 1 called the reluctivity depends on the norm of B. This assumption is simplistically made in most finite-element programs for the calculation of magnetic lenses, for instance, those written by Munro [4] and Lencova [5].

This form, however, is not always sufficient. For instance, it cannot be applied to devices with permanent magnets or with magnetically anisotropic materials. In such cases, we must start from the more general material equation

r=μ0−1Br−MrB,

in which M does not necessarily vanish for B = 0.

The system of basic equations is completed by a relation between the current density j and the electromagnetic field. Its simplest and most familiar form is

=kE,

where k is the conductivity. We shall, however, hardly ever need this equation in charged particle optics, because here we are mainly interested in the spatial distribution j(r), producing a designed magnetic field. The determination of the voltage, to be applied to the coils, is an elementary task.

1.2 ELECTROMAGNETIC POTENTIALS

In the main course of this volume, we shall specialize to stationary, that is, time-independent fields, as these comprise most cases of technical importance; if we have to consider configurations with time-dependent fields, this will be stated explicitly.

1.2.1 Electrostatic Fields

If we disregard the electric field in the coils of magnetic devices, as is usually done, the system of Maxwell’s equations becomes uncoupled, leading to an important simplification. The electrostatic part then reduces to

E=0

ivD=ρ

=εE.

Equation (1.14a) can be integrated once by the introduction of an electrostatic potential V(r),

r=−gradVr.

It is favorable to eliminate the D field completely, whereupon the system (1.14) reduces to one partial differential equation of second order for V(r),

ivεrgradVr=−ρr.

Because ε > 0, this has the basic form of a self-adjoint elliptic equation. In the course of this volume, we shall encounter such a mathematical form frequently in various contexts but with different physical meanings of its variables.

Inhomogeneous dielectric properties are of little importance in charged particle optics (the discontinuity of ε at the surface between different materials must be considered by boundary conditions). With ε = const. Eq. (1.16) simplifies to Poisson’s equation. As this will appear quite frequently, we shall introduce a simplified notation for the Laplace operator,

:=∇≡divgrad.

In cartesian coordinates (and only in these) this operator takes the simple form

=∂2/∂x2+∂2/∂y2+∂2/∂z2.

We shall use the simpler symbol Δ where this is not misleading: otherwise we shall use the notation ∇2. Poisson’s equation now takes the familiar...