are kirchoff laws useless?

[filssavi]

No he is not doing tricks he is considering a physical circuit made of copper and carbon (for resistors) where you have a magnetic field coupling in a closed loop of conductor in other words the middle loop is an antenna that picks up magnetic field from the outside world this induces a voltage in the circuit and hey presto.

That said you could model this effect with a the varing voltage source and only then KCL/KVL could be applied
but that is the electronic equivalent of hiding dirt under the carpet

KVL and KCL are not the laws of conservation!, they are statements derived by laws of conservazione when applied to a specifico set of assumptions (specifically that a lumped element equivalente circuit can be used and that there are no external time varying H or E field in which your physical circuit is immersed)

Conservati in of energy is valid only globally locally can be made not to be true by cleverly picking the boundary ofthe local part in this example if you ignore the energy carried from the magnetic field than yes conservation is broken because you neglected a source of energy

[orolo]

If one looks for a complete breakdown of KCL, it surely must happen at the quantum level, for example, in molecular scale circuits with tiny currents. If a single electron travels along a nanowire, and it reaches a bifurcation, it can propagate along the different wires simultaneously, with different amplitudes, including being reflected back. The electron might tunnel from node to node against the potential, or diffuse along the whole nanocircuit, if it doesn't find anything to interact with. I'm not sure if Ohm's or Kirchoff's laws can be easily reformulated at the operator level, but I have a feeling that KCL at least might, being an statement of conservation (eg. in the example before, the probability current in a node, times charge, must be conserved).