Suppose a resistor is connected on both sides to a copper wire, in a simple circuit with a battery. Resistor and wires are cylinders of equal diameter. A DC current is flowing through the resistor and the wires. The current is axial everywhere, and the electric field is also axial. The strength of the electric field is constant in the resistor, E = U/d (U is voltage, d is length of the resistor), and it is close to zero in the copper wires. This E field seems to be similar to the E field of a parallel plate capacitor. The two boundary surfaces S_{1} and S_{2} between copper and resistor are the source and the sink of the E field. Is it correct to conclude that the boundary surfaces S_{1} and S_{2} carry a surface charge Q = U/C, where C = A ε_{x} /d , and that the resistor behaves like a capacitor for AC frequencies ω > 1/RC ?
That is called parasitic capacitance. Resistors and other circuit components have capacitance. Sometimes (for a given purpose) it is small enough to ignore, other times it must be taken into account. The equations of a circuit are approximations based on assumptions, different approximations are appropriate at different times. What we think of as a resistor, an inductor, or a capacitor is a combination of all of them.
Thanks. I suppose your answer also implies: yes, the boundary surfaces S_{1} and S_{2} carry that surface charge Q. Now, I am trying to understand in mechanical terms why there is a surface charge at S_{1} and S_{2}. DC current through a resistor seems to be mathematically similar to viscous fluid/gas flow through a hydraulic resistor. Is there an equivalent of the electrical surface charge at the entrance of the hydraulic resistor? For example, is the density of the air increased or decreased in a thin layer at the entrance of the air filter in a vacuum cleaner (assuming a constant conduit diameter, to maintain the same particle speed inside and outside the filter)?
I'm not sure that coming up with a 'mechanical' explanation / interpretation will give you any deeper understanding. When you use analogies there is always the risk of coming to conclusions that are 'outside' the overlap between reality and analogy. You seem happy enough with using Maths to describe relationships between variables and Maths is usually an excellent model to use in descriptions of physical 'reality. Why not stick with that? (Unless you are wanting to dumb things down so you can explain to someone else)
Well, I don't trust the results of my Maths. I am unhappy with the idea that there is a uniform surface charge at the boundary layer. A mechanical interpretation might have helped. My question is actually: is Gauss's law valid in a resistor? Or, is the permittivity ε_{x} of conductive/resistive materials infinite? (That would make Q zero).
Oops. In message #1, the formula should have been Q=UC. Hence, the last sentence of message #5 should have been: "Or, is the permittivity ε_{x} of conductive/resistive materials zero? (That would make Q zero)."