In this last section we turn our attention to a question which, despite being somewhat aside from the main route, is extremely important for applications of magnetism in many areas, and especially in medicine. The question relates to the possibility of reconstructing the complete magnetization configuration inside a body, based on measurements of the magnetic field outside it. If this were possible, then we would obtain a powerful tool for studying areas such as current distribution inside a human brain, This in turn would lead to immense progress not only in the diagnosis of a variety of diseases but also to an understanding of how the human brain functions.
From a mathematical viewpoint, we are looking for the solution of an integral Eq. (1.27): provided that the magnetic potential A(r0) outside a body is known everywhere, could we reconstruct the current distribution j (r) inside this body? This would be the same as that of reconstructing the magnetization distribution M(r) from the magnetic field measurements because M(r) can be calculated from the known current distribution using the relationship rot M(r) = j/c (see Section 18.104.22.168) whereby the field can be found as H = rot A.
The problem described above is known as an inverse problem of potential theory (Romanov, 1987) and, unfortunately, cannot be solved uniquely in general. To dem onstrate this fact, we consider first the corresponding problem in electrostatics, namely the reconstruction of charge distribution inside a body from electric field measurements outside it. To show the nonuniqueness of the solution we turn our attention to a simple example, the electric field outside a sphere which carries a total charge Q. It is a well-known text-book result that while the charge distribution inside the sphere remains spherically symmetrical, the field outside the sphere is given by E(r) = Qjr. Hence, this field is exactly the same if, for example: (1) there is a point charge Q in the sphere center; or (2) if the same total charge Q is uniformly distributed on the sphere's surface. Moreover, there is no way to determine the real charge distribution in a sphere unless the field inside the sphere can be measured.
The mathematical reason why such a reconstruction fails is that outside the charged bodies the electric potential satisfies the Laplace Equation Df = 0 (f is a harmonic function). For such functions a so-called Dirichlet problem can be formulated: find a solution of the Laplace Equation Df = 0 outside some closed region W that satisfies some reasonable boundary condition on the surface S of W (f(r e S)= f (r)) and vanishes at the infinity. It can be shown that the solution of this problem is unique - that is, the values of the potential f (r) in the whole space outside some closed surface S can be found if we know its values on this surface f(r e S). Hence, when measuring the potential (or the field) outside a charged body we have at our disposal actually only two-dimensional (2D) information (f-values on any closed surface surrounding a body), which is clearly insufficient to reconstruct a three-dimensional (3D) volume charge distribution inside the body.
The same arguments are valid for the magnetic vector potential. Indeed, this potential satisfies the vector Poisson Equation (1.26) which outside a system of currents (or magnetic samples) transforms into the vector Laplace Equation DA = 0. Again, according to the same solution of the Dirichlet problem, the values of the vector potential in the outer space are completely determined by its values on some closed surface surrounding a system under study, so there is no way to obtain more than 2D information and hence reconstruct a (generally) 3D current or magnetization distribution inside the system.
Although this is a disappointing answer in general, there exist several particular problems for which additional information about the current (magnetization) distribution is available, such that a reconstruction becomes possible. First, in the simplest case when the magnetic field is known to be created by a single pointlike dipole, it is possible to reconstruct its position and the magnitude and orientation of its magnetic moment. In principle, such a reconstruction is possible for any given finite number of dipoles, but in practice its reliability falls rapidly when this number increases.
Another tractable case is when some symmetry properties of the magnetization distribution to be reconstructed are known in advance. If, for example, we know that the magnetization inside a finite cylinder is distributed in axially symmetrical fashion and we can measure the magnetic field on some closed surface surrounding this cylinder, then the reconstruction is (in principle) possible.
In concluding this discussion, we would like to mention that, apart from the principal difficulties demonstrated above, the solution of the Fredholm integral equations of the 1st type f (x)= K(x, y)g (y) dy
(one must solve for g (y) if the left-hand function f (x) and the integral kernel K(x, y) are known) is the so-called ''ill-conditioned problem'' (Press et al., 1992) in the Hadamard sense. This means, that small errors in the experimental data (represented here by f (x)) can cause arbitrary large deviations in the solution if no special precautions (the so-called ''regularization techniques'') are taken. However, this very interesting topic cannot be discuss at this point, and interested readers are referred to literature references in Press et al. (1992).
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