Theory

Assuming the object is non-conductive (so $J = 0$), the relevant Maxwell's equations are

$$\nabla \times H =$$ 0 (1)

$$\nabla \cdot B =$$ 0 (2)

where $B = \mu H$, and the permeability, $\mu$, is related to the susceptibility, $\chi$, by

$$\mu = \mu_0 ( 1 + \chi )$$ (3)

where $\mu_0$ is the permeability of free-space.

These equations can be reduced to a single equation by using the magnetic scalar potential [8] $H = \nabla \phi$. This gives

$$\mu_0 \; \nabla \cdot \left( \, (1 + \chi) \nabla \phi \right) = 0.$$ (4)

Let the susceptibility, $\chi$, be expanded as

$$\chi = \chi_0 + \delta \chi_1$$ (5)

where $\chi_0$ is the susceptibility of air (i.e. $\mu_{\mathrm{air}}= \mu_0 (1 + \chi_0)$ with $\chi_0 = 4 \times 10^{-7}$), and $\delta$ is a constant that represents the average difference in the susceptibility of brain tissue and air (e.g. $-9.5 \times 10^{-6}$ for brain tissues), such that the typical range of $\chi_1$ is from 0 to 1.

Similarly, expand $\phi$ in a series

$$\phi = \phi_0 + \delta \phi_1 + \delta^2 \phi_2 + \ldots$$ (6)

This perturbation expansion in $\delta$ can be substituted back into equation 4 to give

$$\mu_0 ( 1 + \chi_0 ) \nabla^2 \phi_0 =$$ 0 (7)

$$( 1 + \chi_0 ) \nabla^2 \phi_1 + \nabla \cdot ( \chi_1 \nabla \phi_0 ) =$$ 0 (8)

for the zeroth and first order terms in $\delta$.

Using the zeroth order equation together with standard vector calculus identities gives a 3D Poisson equation

$$\nabla^2 \phi_1 = \frac{-1\quad}{1 + \chi_0} \left( \nabla \cdot ( \chi_1 \nabla \phi_0 ) \right).$$ (9)

The Green's function for this equation is

$$G(\mathbf{x}) = \frac{\;-1\quad}{4 \pi r}$$ (10)

where $\mathbf{x}= (x,y,z)$ and $r = \Vert \mathbf{x}\Vert = \sqrt{x^2 + y^2 + z^2}$. This allows the solution of the Poisson equation to be written as a convolution

$$\phi_1(\mathbf{x}) = \iiint G(\mathbf{x}- \mathbf{x}') f(\mathbf{x}') \, d\mathbf{x}'$$ (11)

or more concisely as, $\phi_1 = G * f$, where $f = \frac{-1\;}{1 + \chi_0} \left( \nabla \cdot ( \chi_1 \nabla \phi_0 ) \right)$.

From this the $z$-component of the $B$ field can be written as

$$B_z = \mu H_z = \mu \frac{\partial \phi}{\partial z}$$ (12)

$$= \mu_0 (1 + \chi_0) \frac{\partial \phi_0}{\partial z} + \delta \;... ...ial z} + (1 + \chi_0) \frac{\partial \phi_1}{\partial z} \right) + O(\delta^2).$$

As the zeroth order term is $B^{(0)}_z = \mu_0 (1 + \chi_0) \, {\partial \phi_0}/{\partial z}$, then the first order term is

$$B^{(1)}_z = \frac{\chi_1}{1 + \chi_0} B^{(0)}_z + \mu_0 (1 + \chi_0) \frac{\partial \phi_1}{\partial z}.$$ (13)

Using the fact that

$$\frac{\partial}{\partial x} (G * f) = G * \frac{\partial f}{\partial x} = \frac{\partial G}{\partial x} * f$$

holds for any $G$ and $f$, together with equations 7, 8, 11 and 13, gives

$$B^{(1)}_z = \frac{\chi_1}{1 + \chi_0} B^{(0)}_z - \frac{1}{1 + \c... ...c{\partial^2 G}{\partial z^2} \right) * \left( \chi_1 B^{(0)}_z \right) \right)$$ (14)