## Solution to Maxwells Equations

Maxwell's equations may be solved analytically if the parameters e, and a are constant in space. This ideal case forms the basis of solutions in systems with piecewise constant parameters. The solution to (3.1-3.4) for the fields E and B is obtained by introducing the magnetic (vector) potential A, defined by

and the electric (scalar) potential \$, defined by

Because these equations involve the curl of A and the divergence of \$, and there are vector identities specifying the conditions in which the divergence and curl vanish, there is additional flexibility in defining these potentials. This flexibility is called gauge invariance, and by choosing a convenient gauge:

dt the differential equations for A and \$ separate (Gulrajani 1998).

Assuming harmonic time dependence \$(r,t) = Re[\$(w, t)eiwi], the uncoupled equations have the well-known solutions (Arfken 1995).

These solutions are valid at any frequency, and are therefore useful in electrical impedance tomography (EIT) and transcranial magnetic stimulation (TMS), where the electric and magnetic fields are controlled by an external device that may be driven to high frequencies, e.g., ~100kHz. When applied to EEG and MEG, however, where the frequencies are limited physiologically, these equations may be simplified by the approximation w ^ 0. This is called the quasi-static limit: the fields at each time point t are computed from the sources at that same time point, with no electromagnetic coupling or propagation delays related to the speed of light.