How to prove Kirchhoff's loop rule?

[kelvin490]

In common test books the "proof" is usually use an example of a simple one-loop system with battery and resistor. In such a case, the current actually goes round the loop and the same current flows throughout the loop.

However in actual case, there may be different currents in different sections of the loop, no electron actually returns to the same point.

Another way to use the idea that one point can only have one potential, but are there any proof for that? Why it is not possible to have electrons go through different path to have different energy when they pass through the same point?

 

[berkeman]

It sounds like your understanding and confusion may be more fundamental, but you would prove it in general using induction. Prove it for one loop. Extend that to two loops, and then extend it to n loops...