# Theoretical Study on Geometries of Electrodes in Laboratory Electrical Resistivity Measurement
## 4 Discussion  
### Electrical Resistance Measured between Two Half-Buried Spherical Electrodes  
Herein, we assume two half-buried spherical electrodes, equal in radius (\(r\)) and quantity of charge but opposite in polarity, are separated by the distance \(L\) (Figure 5). The electric potential of an arbitrary point, situated at a distance \(s\) from the electrodes, is obtained from the equipotential surface area (\(A(s)\)):  
\[V_{5}=-V_{6}=\int\limits_{\tau}^{L-r}\frac{\rho I}{A(s)}ds=\int\limits_{\tau}^ {L-r}\frac{\rho I\,ds}{\underbrace{2\pi s^{2}}_{\text{surface area of}}}=\frac{\rho I}{2\pi}\left(\frac{1}{r}- \frac{1}{L-r}\right) \tag{14}\]  
where \(V_{5}\) is the electric potential from the positive-charge electrode and \(V_{6}\) is the electric potential from the negative-charge electrode. \(I\) is the electric current between the two electrodes, and \(\rho\) is the electrical resistivity of the material. The equipotential surface area of the half-spherical electrodes is identical to that of the point electrodes in the previous section.  
Through [PERSON]'s law, the electrical resistance between two half-spherical electrodes (\(R_{half-sphere}\)) is:  
\[R_{half-sphere}=\frac{V_{5}-V_{6}}{I}=\frac{\rho}{\pi}\left(\frac{1}{r}-\frac {1}{L-r}\right) \tag{15}\]  
This is identical to Equation (10) and the 3D resistance reported in Reference [19]. If the distance between the two electrodes increases, the electrical resistance between them continuously increases (the derivative of \(R_{half-sphere}\) about \(L\) is always positive) and finally becomes \(\rho/\pi r\), as reported by many researchers [20; 21]. Additionally, the electrical resistance becomes zero when the two electrodes are in contact with each other (\(L=2r\)).  
Even though the cylindrical electrodes are commonly utilized, not all of the electrodes for the electrical resistivity measurement are cylindrically shaped [22; 23; 24; 25]. The concept of equivalent electrodes is revisited and commonly adopted for a simple calculation of the shaped electrodes. The shapes of the equivalent electrodes become half-spheres which have the same surface area as the original electrodes. The radius of the equivalent electrodes (\(r_{e}\)), for the cylindrical electrodes with spherical tips, can be seen as  
\[2\pi r_{e}^{2}=2\pi rl+2\pi r^{2}\Rightarrow r_{e}=r\sqrt{1+\frac{l}{r}} \tag{16}\]  
\[R_{equivalent}/R_{cylindrical}=\frac{l}{r}\left(\frac{1}{\sqrt{1+\frac{l}{r}}}- \frac{1}{\frac{l}{r}-\sqrt{1+\frac{l}{r}}}\right)\left[\ln\left(1+\frac{l}{r} \right)-\ln\left(1+\frac{\frac{l}{r}}{\frac{l}{r}-1}\right)\right] \tag{17}\]  
Figure 5: Definitions of the symbols and equipotential surface shape between two half-buried spherical electrodes.  
Figure 6 displays the electrical resistance between two cylindrical electrodes (Equation (8), solid line) and between two equivalent electrodes (Equation (15) using Equation (16), dashed line) in a brine with an electrical resistivity of 4.78 \(\Omega\)m. It is trivial that the radius of the equivalent electrode is bigger than that of the cylindrical electrode from Equation (16). Therefore, the distance between electrodes that makes the electrical resistance zero in Figure 6 is slightly larger in the case of the equivalent electrodes. The gap between the solid and dashed line decreases as the ratio between the penetration depth and the radius (\(l\)/\(r\)) decreases and increases when \(l\)/\(r\) increases and converges. Figure 7 depicts the relationship between \(L\)/\(r\), \(l\)/\(r\), and \(R_{equivalent}\)/\(R_{cylindrical}\) using Equation (17). If one were to treat the cylindrical electrodes as half-buried spherical electrodes, Figure 6 could be used as a guideline for determining the allowable error. For instance, if the allowed error is 2% (0.98-1.02 in \(y\)-axis of Figure 7) and \(l\)/\(r\) is determined as 2, \(L\)/\(r\) should exist from 8 to 11.