I'm reading through my E&M textbook (Physics for Scientists and Engineers 3rd edition, Knight) and watching as many lectures on YouTube (Shankar, Pomerantz, Lewin, etc) to prepare for next quarter. I have one question as I don't want to learn a wrong concept.
I've noticed when you charge a capacitor with a constant current you get a voltage slope. After watching Carver Mead talk on G4g, I later saw that for an inductor $V = L \frac{dI}{dt}$. Which looks strikingly similar to the equation I've learned up to this point for capacitors, $I = C \frac{dV}{dt}$. That's weird.
If you charge a capacitor with a constant current you get a slope of the voltage, then do you get the slope of the current if you charge an inductor at a constant voltage?
Are these mirrored concepts?
If this is the case it makes sense to me. So much of an electrons motion seems dependent on it's path or type of path. At least from what I've learned to this point. Conductors and inductors seem to be taking advantage of the same rules for different outcomes.
Answer
Capacitors and inductors are images of one another under the self-inverse mapping that transforms a linear electrical network to its dual network.
The network duality transformation maps the network's graph to its topological dual graph,then all the impedances (either as lone-frequency complex scalars or as Laplace transfer functions) in the dual graph links become their reciprocals and current sources become voltage sources and contrariwise.
The physical meaning of this mapping is that we are finding a network where the voltages and currents in the network's state equations swap roles. So this is the reason for your observation: your two equations result from one another if you swap the roles of the voltage and current.
Some common examples: the Norton equivalent source is the dual of the Thévenin equivalent source and contrariwise. Likewise the star-delta transformation is an evocative example, showing how loops become links and contrariwise in the dual graph.
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