A two dimensional sheet with a uniform sheet conductivity of σ has a central metallic point contact and a circular metal contact at the boundary as shown in the figure.
If a constant current I is injected through the central contact and collected at the boundary, then the voltage difference between two points on the sheet at radius r1 and r2 is proportional to
1
\(\frac{I}{\sigma}\left[\tan ^{-1}\left(\frac{r_2}{r_1}\right)-\frac{\pi}{4}\right]\)
2
\(\frac{I}{\sigma}\left[\ln \left(\frac{r_2}{r_1}\right)\right]\)
3
\(\frac{I}{\sigma}\left(\frac{r_2-r_1}{r_2+r_1}\right)\)
4
\(\frac{I}{\sigma}\left(\frac{r_2-r_1}{r_2+r_1}\right)^3\)