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Consider $n$ conductors $L1,L2,\dots $ in space containing the charge $Q1,Q2,\dots$


The scalar Potential $V1,V2,\dots$ on each conductor is constant. The potential is proportional to the chargedensity on the conductor \begin{align} V_i=\sum_{j=1}^n p_{ij} Q_j \quad (i=1,2, \dots n) \end{align} where the $p_{ij}'s$ depend only on the geometry. The inverse relation is given by \begin{align} Q_i=\sum_{j=1}^n C_{ij} V_j \quad (i=1,2, \dots n) \end{align} where $C_{ii}$ are the capacities and the $C_{ij}$'s for $i\not=j$ are called induction coefficients. The capacity of a conductor is therefore equal to the total charge on the conductor, when held at fixed potential $V_i=1$ and all other conductors held at zero potential.

The potential energy of a conductorsystem is \begin{align} W=\frac{1}{2} \sum_{j=1}^n Q_iV_i=\sum_{j=1}^n C_{ij}V_iV_j \end{align}

For applications to more complicated problems, Variational Calculus can be employed to get a approximate solution (see Further Reading).

Further Reading:

  • John David Jackson - Klassische Elektrodynamik