# Relation between potential and charge of a group of conductors

studying electrodynamics I encountered a few weeks ago this statement regarding a set of conductors in space with no free charges:

I could not find an explicit proof of this in any book and I did not manage to prove it myself. More specifically, another book states that, if all the total charges of conductors are zero except one, then the potentials are proportional to this total charge $$V'_1=p_{11}Q_1 \qquad V'_2=p_{21}Q_1$$(I'm doing this with two conductors for simplicity) and viceversa, if the total charge on the the first conductor is zero and the charge on the second is $$Q_2$$ the potentials are $$V''_1=p_{12}Q_2 \qquad V''_2=p_{22}Q_2$$ and, if both conductors are charged then the potential is $$V_1=p_{11}Q_1+p_{12}Q_2 \qquad V_2=p_{21}Q_1+p_{22}Q_2$$So, he makes a statement even stronger than the one in the image above and the only justification they give is that the equations on potential and electric field are linear.

Can someone post a proof of this or just tell me where can I find one?

• Is this a very difficult thing or am I missing something? – Rhino yesterday