## 3 Results  
When we calculate the voltages, we find that \(\psi\) and \(V\) are nearly the same in the wire and the tissue: they are both approximately linear in \(x\) (Figure 2), and both are nearly sinusoidal in time (Figure 3). This behavior occurs because the diffusion time (\(L^{2}/D=0.56~{}\mathrm{\SIUnitSymbolMicro s}\)) is much less than the period of the applied current (333 \(\mathrm{\SIUnitSymbolMicro s}\)). The voltage difference between the ends of the wire, \(\Delta V_{i}\), is nearly equal to \(\Delta\psi\) for \(r_{e}>>r_{i}\), so from Figure 2, we see that the voltage difference is about 1.7 mV. The voltage in the tissue, \(V_{e}\), is approximately the difference between \(\psi\) and \(V\), which is difficult to determine from Figures 2 and 3 but is not zero (Figure 4). It has a maximum amplitude at the ends of the coil and a maximum slope at the center. The magnitude of the electric field in the tissue is about 4 mV/m at \(x=0\). \begin{table}
\begin{tabular}{c c c c} \hline
**Parameters** & **Definition** & **Value** & **Unit** \\ \hline \(\kappa\) & Insulation dielectric constant & 3.8 & - \\ \(\sigma_{e}\) & Tissue conductivity & 0.1 & \((\Omega\mathrm{m})^{-1}\) \\ \(\sigma_{i}\) & Wire conductivity & \(60\times 10^{6}\) & \((\Omega\mathrm{m})^{-1}\) \\ \(d\) & Insulation thickness & 300 & nm \\ \(a\) & Wire radius & 2.5 & \(\mathrm{\SIUnitSymbolMicro m}\) \\ \(b\) & Tissue radius & 0.1 & mm \\ \(L\) & Half the wire length & 1 & mm \\ \(I_{o}\) & Amplitude of applied current & 1 & mA \\ \(f\) & Frequency & 3 & kHz \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters used in the calculation (from [PERSON] et al. [18]). Figure 2: (**a**) The weighted average voltage \(\psi\) and (**b**) the voltage across the insulation \(V\) as functions of position \(x\) at time \(t=83~{}\mathrm{\SIUnitSymbolMicro s}\) (the time of the peak of the first phase of the sine wave). The time course for \(V_{e}\) and \(E\) (Figure 5) are approximately proportional to \(\cos(\omega t)\). Therefore, the tissue electric field is 90 degrees out of phase with the coil current. Because the initial condition is \(V_{e}=0\), the tissue potential rises abruptly near \(t=0\) (Figure 6). In about one microsecond (roughly the diffusion time) it reaches its asymptotic behavior. Figure 4: (**a**) The voltage in the tissue \(V_{e}\) and (**b**) the tissue electric field \(E\) as functions of position \(x\) at time \(t=333\) μs. Figure 5: (**a**) The voltage in the tissue, \(V_{e}\), at position \(x=-10\) mm and (**b**) the tissue electric field, \(E\), at position \(x=0\), both as functions of time \(t\). Figure 3: (**a**) The weighted average voltage \(\psi\) and (**b**) the voltage across the insulation \(V\) as functions of time \(t\) at position \(x=-10\) mm (the left end of the wire). The voltage drop across the tissue is about 0.005 mV, compared to a drop of 1.7 mV in the wire. The tissue voltage drop, and therefore the tissue electric field, is so small because of the high resistance of the current path through the tissue (about 640,000 \(\Omega\)) compared to that through the wire (1.7 \(\Omega\)). Neither the tissue conductivity nor the radius of the cylindrical tissue space is known accurately, and the tissue conductivity may be heterogeneous [25]. Figure 7 shows the peak electric field strength as a function of the tissue resistance per unit length, \(r_{e}\). [PERSON] et al. [18] performed a test to determine if capacitive currents were important by passing a large transient current to burn a small portion of the coil, leaving an open circuit. We can simulate this experiment by dramatically increasing \(r_{i}\). When we set \(r_{i}=30\) M\(\Omega\)/m, we found that the electric field was 143,385 mV/m. The stability criterion defined earlier slows the numerical calculation because it implies a very small time step. We had to use a time step (0.001 \(\upmu\)s) that is more than 100,000 times shorter than the period of the stimulation current (333 \(\upmu\)s).