Power dissipation in parallel circuit

[BruceW]

But yeah, it makes perfect sense. Let's say the circuit runs for some time τ. Some quantity of charge q has flown through it. The total energy dissipated is ∫Vdq = ∫Pdt. What would it mean for power to be unoptimized? It would mean that one of these elements dq would be able to find a path through your circuit with lower V. That would mean that different branches have different voltages. So the solution must certainly be a minimum in P in a steady state circuit.

I'm sorry, but I don't fully understand what you're saying here. It sounds (to me) like you're saying that unoptimised power means that there is more than one possible value for the power dissipated by a charge element going through the circuit. But I don't see why this is necessarily true?

And for clarity to anyone who was wondering, the optimisation of power which I wrote in my first post is done by keeping the resistances constant, and the total current constant. And varying the currents through each resistor, while keeping to the constraint that the currents 'add up' at a node.

I've been thinking about this for aaages, but I still can't think of the reason behind why this happens.

 

[Tipox]

It is interesting to notice this method works only with resistors.
It does not work with a circuit containing diodes and resistors !
Such a method would lead to a minimized dissipated power but voltages are not the same between to voltage nodes (try it with a diode and a resistor in parallel!).
I am curious to find out why...

If somebody has an idea...