3.2 Capacitance of N Conductors

Given a group of n conductors carrying linear charge densities q1, q2, …, qn that are located above the ground plane, equations of the same form as Equation 31 (Section 3.1) can be constructed for all conductors in the group. Expressing the complete set of n potential equations in matrix notation yields

𝐕=𝐏𝐐\mathbf{V=PQ} (32)


V is the voltage vector.

Q is the charge vector.

P is the potential coefficient matrix.

The elements of the potential matrix (with units of F-1m) are defined as follows:

pij={ln(Diidi)2πϵifi=jln(Diidij)2πϵifij{p}_{ij}=\begin{cases}\frac{ln\left(\frac{D_{ii}}{d_{i}}\right)}{2\pi\epsilon}% &\operatorname{if}i=j\\ \frac{ln\left(\frac{D_{ii}}{d_{ij}}\right)}{2\pi\epsilon}&\operatorname{if}i% \neq j\end{cases} (33)

Recall that the permittivity of a medium is often expressed as

ϵ=ϵ0ϵr\epsilon=\epsilon_{0}\epsilon_{r} (34)


ϵ0\epsilon_{0} is the permittivity of free space (i.e. 8.8541853×10-12F/m).

ϵr\epsilon_{r} is the relative permittivity of the medium (e.g. 1 for air).

In matrix notation, the capacitance of the configuration is

𝐐=𝐂𝐕\mathbf{Q=CV} (35)

solving Equation 32 for the charge vector yields

𝐐=𝐏-𝟏𝐕\mathbf{Q={P}^{-1}V} (36)

By inspection it is apparent that

𝐂=𝐏-𝟏\mathbf{C=P^{-1}} (37)

The matrix C is sometimes known as the capacitance coefficients (or Maxwell’s coefficients) of the line.