Why is kirchoofs law applied at small temperature range?
Answers
When you say Ohms law as V=IR, it is actually a model. A model is a framework designed to provide answers using calculations. During the modeling process we approximate the parameters for convenience and mathematical handling. One such approximation in V=IR is the consideration of lumped parameters - resistance is concentrated at a point. Till the real scenarios are close to this, like manufactured resistors or small circuit resistances, the equation will give correct answers. But here is a catch, how small or local should the resistances be? Answer is that it is relative to the signals - voltages and currents - used. A high frequency signal will have small wavelength - spread in the space, while lower frequency will have large spread. The distances the resistances is spread is compared with this wavelengths. Thus there is no absolute threshold of lumpedness it's relative to frequencies.
In my MS thesis, I simulated a medium for electromagnetic wave using transmission line model. These are basically L and C circuits in series. Here L and C are lumped approximations to magnetic permeability and electric permittivity respectively. My results perfectly matched the theory I was testing. But, when I started playing with the input frequency I noticed there are distortions in my output proportional to frequency used. It took me some time to decipher the cause, and in the process I learned an important thing which I have explained earlier - as the wavelength of signal shrink the lumpedness becomes comparable to the signal size.
I could reduce distortion by adjusting the L and C values per length of the medium/transmission line. This means any model which uses parameter per space and even the signal values not as absolute but relative to space like current per length or per area (current density - J) or voltage/length (which is E) will work as both signals and parameters are adjusted according to each other. This is why your second equation works, but not the first one.