The definition of mutual capacitance

Question

I am not sure I completely understand the definition of mutual capacitance. Let's say we have two conductors, $A$ and $B$, so that the following holds:

1. Both conductors are isolated.
2. $A$ is isolated and $B$ is kept at fixed potential.
3. Both conductors are kept at fixed potential.

What is the correct definition (and meaning) of (mutual) capacitance in these three cases? For example, in the first case, we could have arbitrary charges $Q_A$ and $Q_B (\neq - Q_A)$ on the conductors. These charges would result in the potentials $V_A$ and $V_B$. Now, do we define capacitance as $$C = \frac{\Delta Q}{\Delta V},$$ where $\Delta V = V_A - V_B$ and $\Delta Q$ is the charge which would be transferred from $A$ to $B$ if we connected the conductors by a thin wire? If this is correct, the definition still makes sense in the second example but completely fails in the third one. Does it even makes sense to define the capacitance for the system in the third example?

Also, is there a easy way to determine the mutual capacitance from capacitance matrix?

Answer 1

Suppose you set the zero of potential so both conductors have zero charge at zero potential. If you then set them to potentials $V_A$ and $V_B$, you can prove that they will acquire charges \begin{align} Q_A=C_A V_A+C_{AB}V_B,\\ Q_B=C_{BA}V_A+C_B V_B, \end{align} respectively. This is the real definition of capacitance, and particularly of mutual capacitance: the coefficients in the linear multi-variable relations between the charges on the conductors and the corresponding potentials; the mutual capacitances are the off-diagonal elements of this capacitance matrix.

Whether the charges and potentials are achieved by, say,

• grounding the potentials,

• connecting them to ground via a potential source like a battery, or

• isolating them and injecting a specific amount of charge,

does not really change anything - it simply tells you which terms you hold constant and which ones are variables that are determined by the geometry of the situation. In particular, note that if I am allowed to set $A$ and $B$ to fixed (but arbitrary) potentials, then I am really in your case 3. Here $C_{AB}$ is easily seen to be the increase in $Q_A$ per unit increase in $V_B$ while $V_A$ is kept fixed - nothing mysterious about it, and no hint of undefiniteness around. The definition of the mutual capacitance does not depend on the situation - only on the geometry.