Analytical Sphere

The analytical magnetic field (including Lorentz Correction) produced by a spherical object of radius $R_0$ and susceptibility $\chi_{i}$ inside a medium of susceptibility $\chi_{e}$ is given by

$$B_z = \left\{ \begin{array}{lccl} B_0 + \frac{\chi_{i} - \chi_{e}... ...uad & \mathrm{for} & r>R_0 \\ B_0 & & \mathrm{for} & r<R_0 \end{array} \right.$$ (28)

where cylindrical coordinates are used with $\theta$ being the angle to the positive $z$ axis.

Figure 1: Sphere object model (left), analytical field (middle) and perturbation field calculation (right) for two sizes of sphere: $R_0=64$ (top row) and $R_0=8$ (bottom row).

Figure 2: Plots of the $B_z$ distribution along the z axis for spherical objects of radius $R_0 = 2, 4, 8, 16, 32, 64$ voxels. Results are shown for the analytical calculation (theory), the perturbation calculation and the difference between the two. Note that the values are only calculated at the centres of the voxels -- hence the noticeably non-vertical sides in the $R_0=2$ case.

Figure 1 show the field distribution (less $\mathrm{B}_0$) along the $z$ axis of a sphere, comparing the analytical solution given in equation 28 with the solution calculated with the perturbation method. Figure 2 shows plots through the centres of these spheres for a range of different radii, $R_0 = 2, 4, 8, 16, 32, 64$ mm, where in each case the voxel size is $1 \times 1 \times 1$ mm. This demonstrates the relationship between spatial extent and size of the error, where it can be seen that the maximum error is approximately equal to 1 ppm in all cases, with the spatial extent of significant errors covering 5 voxels or so. These results are very similar to those shown in [4,1,2,5] despite the range of different object models (e.g. boundary element methods vs voxels) and approximations used.