What is the path electricity have to travel after I turn on a switch until it reaches the device? My question is not about the speed of electricity, but about the distance it have to travel after being switched on, which I call "the path".
A lot of people say that the path is simply from the switch to the device. But I'm wondering, isn't an open circuit completely dead? So before the switch was turned on, there should be no electricity in the part of the wire between the power source and the switch - any more than in the part of the wire between the switch and the device. So the path should be from the power source to the device.
But how should the power source "know" that the circuit is completed? Probably there is a need for a signal to travel from the switch to the power source. But it doesn't make sense that the switch is capable of sending signals, so the signal should come from the other end of the power source (let's say the negative, to talk from DC). Then we are now back to the beginning. It's paradoxical.
 A: Transmission line effects is the label to encapsulate the behaviour you are talking about. At least in EE.
In an electric circuit, there are transient signals bouncing back and forth between all parts of the circuits as the different parts of the circuit "communicate and figure out" what state they should settle on. When you hear about signals ringing, overshooting, or undershooting, this is because of that.
A mechanical analogy would be that when you dump water into a pipe, how do you know the other end isn't a dead end? Or gets narrower? The answer is you don't until the water reaches that end and the pressure wave reflects back to splash you in the face. Then you know that you need to reduce the water flow to accommodate the narrowing, or stop it completely to accommodate the blockage. The splash is like overshooting.
With electricity this might happen very fast, so fast that in many cases you don't need to worry about but it does happen. Much more noticeable with extremely long cables or very high frequencies.
Part of the transmission line are parasitic capacitances in space between the two lines that the electric field can interact through through space. You could say that the initial current flows through that as it progresses down the line before it is established as actually flowing through the entire line. I believe physicists call this the "displacement current".


You can try watching this and see if you understand it:
https://www.youtube.com/watch?v=iph500cPK28&list=PLlD-hnyodqSf7D9zPVviYaOaDZnWxkDYx&index=28
