"Scalar Potential" Approach
2.1 INTRODUCTION: A MODEL PROBLEM
Let us now tackle problem (31) from Chapter 1: magnetostatics. We
need a model problem for this discussion; we need it to be as simple as
possible, and still come from the real world.
The following, known as the "Bath cube" problem [DB], will do. It is
concerned with a device, built around 1979 at Bath University, which
was essentially a hollow box between the poles of a large electromagnet
(Fig. 2.1). In this almost closed experimental volume, various conducting
or magnetizable objects could be placed, and probes could be installed to
measure the field. The purpose was to confront what computational
codes would predict with what these probes recorded. The problem was
one in a series of such benchmark problems, regularly discussed in an
ad-hoc forum (the TEAM international workshop [T&]). Comparative
results for this one (known as "Problem 5") can be found in [B5].
2.1. The "Bath cube" benchmark. Both coils bear the same intensity I.
The magnetic circuit M is made of laminated iron, with high permeability
(~t > 1000 ~t0). Various objects can be placed in the central experimental space.
32 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach
Problem 5 was actually an eddy-current problem, with alternating
current in both coils, and we shall address it in Chapter 8. What we
discuss here is the corresponding static problem, with DC currents: given
the coil-current, find the field inside the box.
It will be some time before we are in a position to actually solve this
problem, despite its obvious simplicity. For before solving it, we must
set it properly. We have a physical situation on the one hand, with a
description (dimensions, values of physical parameters) and a query (more
likely, an endless series of queries) about this situation, coming from
some interested party (the Engineer, the Experimenter, the Customer,
...). To be definite about that here, we shall suppose the main query is,
"What is the reluctance of the above device?" The task of our party (the
would-be Mathematical Modeller, Computer Scientist, and Expert in the
Manipulation of Electromagnetic Software Systems) is to formulate a
relevant mathematical problem, liable to approximate solution (usually
with a computer), and this solution should be in such final form that the
query is answered, possibly with some error or uncertainty, but within a
controlled and predictable margin. (Error bounds would be ideal.)
Mathematical modelling is the process by which such a correspondence
between a physical situation and a mathematical problem is established. 1
In this chapter, a model for the above situation will be built, based on the
so-called "scalar potential variational formulation". We shall spiral from
crude attempts to set a model to refined ones, via criticism of such attempts.
Some points about modelling will be made along the way, but most of
the effort will be spent on sharpening the mathematical tools.
First attempt, based on a literal reading of Eqs. (1.31). We are given a
scalar field ~t, equal to ~ in the air region, and a time-independent
vector field j (actually, the jg of (1.31), but we may dispense with the
superscript g here). From this data, find vector fields b and h such that
(1) rot h = j,
(2) b= ~th,
(3) div b = 0,
in all space.
The first remark, predictable as it was, may still come as a shock:
This formulation doesn't really make sense; the problem is not properly posed
1 It requires from both parties a lot of give and take.