I have been thinking about Ohm's Law in one particular situation. say I have a trace on top of a ground plane on a pcb. A signal is driven from point A and terminated by a load at point B. Looking at the return current path using Ohm's law V=IR. This implies most of the current should be running in a path of least resistance on the ground plane, which is the straight line from point B back to point A. This is because any path deviated from the straight line from B to A is a longer path, whereby, have higher resistance. But we all know signal is travel as EM wave, the trace and the ground plane form a microstrip guided structure. If the signal from A is of high frequency ( over say 50KHz give and take), the ground return path pretty much follow the trace and most current concentrated right under the trace. So if I snake the trace around while going from point A to B, the current is not following the path of least resistance anymore. In fact the current follow the path of least impedance as the guided structure will have the lowest characteristic impedance when the return path is closest to the forward path. How does Ohm's Law explain this. I can see Ohm's Law will work in "point form" at any point, but in macro situation, electromagnetic theory take over. Please correct me if my assumption is wrong.
Here is the drawing. The one on the left is the return current density on the ground plane from B to A because the path of lowest resistance is the straight line from B to A. This is what Ohm's Law implies....that the highest current is when the resistance is the lowest at a given voltage. Of cause you expect a distribution that spread out as a potential can form between B and A as current increase and make the round about path starting to conduct some current. But the highest current density should be on the straight line from B to A. On the right, the drawing show the trace on the pcb above the ground plane that "snake around from B to A. The trace with the ground under forms a microstrip structure that EM propagate from B to A. The return image current follow the trace closely on the ground plane right below the trace. So the image current really snake along with the trace on top. It is know that 95% of the current will be in the narrow path a few trace width of the trace on top. So this no longer follow the path of least resistance, but infact, it follow the path of least impedance where the lowest impedance of the microstrip is form between the trace on top to the grounding plane right under the trace. Any path on the ground plane away from the trace will form a higher impedance structure and also have larger area enclosed by the forward and return current path. Here are two articles that explain how the image current on the ground plane flows. http://paginas.fe.up.pt/~hmiranda/etele/microstrip_basics.pdf http://www.ediss-electric.com/technical_pdf/currentpath.pdf
If the material is homogeneous. This looks a bit like the skin effect. Magnetic fields, nonlinear material response and probably some other things can give deviations from Ohm's law. For a high-frequency current, magnetic fields can be relevant.
So there is limitation in Ohm's Law, I am not imagining things? Skin effect come into play.....even at surprising low frequency of a few hundred kilo hertz. But this is more about the structure is a guide structure for EM wave to travel in.
Of course there is a limitation of "Ohm's law" as it's a simplification of EM field theory that's usually true only when the results are independent of time. The trace and ground plane charge carriers (electrons) move only microscopic distances when affected by the changing fields of the signal moving from point A to B at almost light speed. So AC current flow distribution on the ground plane is tied to the strength of the field at that point (proximity effect) and how far that charge moves under that field (drift velocity) during the time that field exists. http://www.polarinstruments.com/support/cits/AP174.html
I guess i missed something. Well of course it does and ohm says so I = E/Z, Z is a function of frequency. Maybe some other things, too, like Levin's induction. i think i'm too shallow for this one.
But the path has nothing to do with current, it is the impedance of the wave guide for the EM wave to travel. Current is only the consequence of the boundary condition of EM wave. In fact, the voltage on the tx line is from the EM wave as [tex]V=-\int_c \vec E\cdot d\vec l[/tex] Yes, in point form of Ohm's Law for every single point along the tx line, Ohm's Law work as V=IZ. But both I and V along the tx line are all from the effect of the EM wave. All these don't follow the true current and voltage interpretation that voltage across point A and B cause the most current to flow in the path of least resistance on the ground plane.