# Theoretical Study on Geometries of Electrodes in Laboratory Electrical Resistivity Measurement
## 2 Theoretical Background
### Two Cylindrical Electrodes in the Container  
The method of image charges has been employed to describe electrical and magnetic fields with even and specified boundary conditions. The underlying theorem that explains the uniqueness of the solution facilitates solving a cumbersome Laplace equation. Figure 2 shows the equivalent system by replacing the non-conductive wall of the container embedded with two cylindrical electrodes into image charges of both side-walls. Note that potentials at the side-walls become zero. In the original  
Figure 1: Definitions of the symbols and equipotential surface shape between two cylindrical electrodes.  
system, the current, \(I\), flows from electrode \(M\) to electrode \(N\), and the potential difference between electrode \(M\) and electrode \(N\) in the container can be calculated from Equation (8) and potentials from the imaginary electrodes involved in the electrode geometry and spacing.  
\[V_{3}=\frac{\rho I}{2\pi l}\left\{\begin{array}{c}\underbrace{\ln \left(1+\frac{l}{r}\right)}_{\text{effect of M}}+\underbrace{\left(-\ln\left(1+ \frac{l}{L-r}\right)\right)}_{\text{effect of N}}\\ +\left[\underbrace{\ln\left(1+\frac{l}{W-L-r}\right)}_{\text{effect of N}^{\prime}}+ \underbrace{\left(-\ln\left(1+\frac{l}{W-r}\right)\right)}_{\text{effect of M}^{\prime\prime}}+\underbrace{\text{ effect of N}^{\prime\prime}}_{\text{effect of N}^{\prime\prime}}\\ +\left[\underbrace{\ln\left(1+\frac{l}{W-L-r}\right)}_{\text{effect of container wall}}+\left(-\ln\left(1+\frac{l}{W-r}\right)\right)\right]\\ \end{array}\right\}\right\} \tag{11}\] \[V_{4}=\frac{\rho I}{2\pi l}\left\{\begin{array}{c}\underbrace{ -\ln\left(1+\frac{l}{r}\right)}_{\text{effect of N}}+\underbrace{\ln\left(1+ \frac{l}{L-r}\right)}_{\text{effect of M}}\\ +\left[\underbrace{-\ln\left(1+\frac{l}{W+L-r}\right)}_{\text{effect of M}^{ \prime}}+\underbrace{\text{effect of N}^{\prime\prime}}_{\text{effect of M}^{\prime\prime}}+\underbrace{\text{ effect of M}^{\prime\prime}}_{\text{effect of M}^{\prime\prime}}\\ +\left[\underbrace{-\ln\left(1+\frac{l}{W+L-r}\right)}_{\text{effect of container wall}}+\left(-\ln\left(1+\frac{l}{W-L-r}\right)\right)+\ln\left(1+\frac{l}{W-r} \right)\right]\\ \end{array}\right\}\right\} \tag{12}\]  
The theoretical resistance of the cylindrical electrodes in a non-conductive container is:  
\[R_{boundary}=\frac{V_{3}-V_{4}}{I}=\frac{\rho}{\pi l}\left\{\begin{array}{c} \underbrace{\ln\left(1+\frac{l}{r}\right)}_{\text{effect of an electrode}}+ \underbrace{\left(-\ln\left(1+\frac{l}{L-r}\right)\right)}_{\text{effect of the other electrode}}\\ +\left[\underbrace{\ln\left(1+\frac{l}{W+L-r}\right)}_{\text{effect of container wall}}+\ln\left(1+\frac{l}{W-L-r}\right)-2\ln\left(1+\frac{l}{W-r}\right) \right]\\ \end{array}\right\}\right\} \tag{13}\]  
Figure 2: Basic concept of imaginary electrodes.  
where \(W\) is the width of the container.  
Only two boundaries vertical, to the electrodes' movement direction, are considered in Equation (13) because the distance to the other boundaries is constant and far enough from the electrodes. In addition, the electrical resistivity of the boundaries is assumed as an infinite since the electrical resistivity of the acrylic is much higher than that of the brine.