Quasi-Static Approximation

EIT systems inject alternating currents at frequencies from roughly 1 kHz to 1 MHz. At these frequencies the wavelength of electromagnetic waves is far larger than the body, and magnetic induction is negligible. Maxwell’s equations then reduce to their quasi-static form:

• $\nabla \times \mathbf{E}\approx 0$, so $\mathbf{E}=-\nabla u$ for a scalar potential $u$;
• the displacement and conduction currents together are divergence-free.

With time-harmonic excitation $e^{i\omega t}$, the conductivity is replaced by the complex admittivity $\gamma =\sigma +i\omega \epsilon$ (see Complex Conductivity). The potential then solves $\nabla \cdot (\gamma \nabla u)=0$ with complex coefficients.

In the DC (or low-frequency) limit $\omega \epsilon \ll \sigma$, the admittivity is real and the model reduces to the real Conductivity Equation. This is the setting of most of this wiki.

References

1. M. Cheney, D. Isaacson, J. C. Newell (1999). Electrical Impedance Tomography. SIAM Review 41(1), 85–101. doi:10.1137/S0036144598333613
2. E. Somersalo, M. Cheney, D. Isaacson (1992). Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography. SIAM J. Appl. Math. 52(4), 1023–1040. doi:10.1137/0152060