Magnetic Fields of a Dipole in Special Volume Conductor Shapes

Expressions are presented for the magnetic fields produced by current dipoles in four basic volume conductor shapes. These shapes are the semi-infinite volume, the sphere, the prolate spheroid (egg-shape), and the oblate spheroid (discus-shape). The latter three shapes approximate the shape of the human head and can serve as a basis for understanding the measurements of the brain's magnetic fields. The semi-infinite volume is included in order to investigate the effect of the simplest boundary between a conductor and nonconductor. The expressions for the fields are presented in a form which separates the total field into two parts. One part is due to the dipole alone (the dipole field); the other is due to the current generated in the volume conductor by the dipole (the volume current field). Representative plots of the total field and the volume current field are presented for each shape. The results show that for these shapes the component of the total field normal to the surface of the volume conductor is produced completely or in large part by the dipole alone. Therefore, measurement and use of this component will greatly reduce the complexity of determining the sources of electrical activity inside a body from measurements outside the body by removing the necessity of dealing with the volume current field.