I agree with you abut Ohm's, law that it's just a ratio. It probably started at its inception as general rule for different materials and retained its name for historical reasons as more and more "non-Ohmic" materials were discovered.
sophiecentaur said:
And, btw, KVL and KCL are different cases. They are Identities, rather than Laws
Here I beg to differ. I would not call them identities but consequences of equations that are based on empirical evidence, laws if you wish.
KVL is a direct consequence of Faraday's "law" in the static case as expressed by Maxwell's equation for the curl of the electric field. It is essentially an energy conservation equation because the electric field is conservative. Formally, starting with Faraday's law when there is no time-varying magnetic field, $$ \mathbf{\nabla}\times\mathbf{E}=0\implies \oint \mathbf{E}\cdot d\mathbf{l}=0\implies -\sum_i\Delta V_i=0.$$
KCL is a result of the "law" of charge conservation which is written as $$\mathbf{\nabla} \cdot \mathbf{J}+\frac{\partial \rho}{\partial t}=0$$ Construct a closed Gaussian surface enclosing any part of the circuit, multiply by a volume element ##dV## and integrate over the volume to get $$\int_V \mathbf{\nabla} \cdot \mathbf{J}~dV+\int_V \frac{\partial \rho}{\partial t}~dV=0.$$ Use the divergence theorem to transform the first volume integral into a surface integral and rearrange to get $$\int_S \mathbf{J}\cdot~\mathbf{\hat n}~dA=- \frac{\partial}{\partial t}\int_V\rho~ dV.$$The left hand side represents the total current coming
out of the surface of volume ##V## because ##\mathbf{\hat n}## is the outward normal. If there are ##N## locations on the surface where current comes out, the left hand side can be written as $$I_{\rm{out}}=\sum_{i=1}^{N}\int_{S_i} \mathbf{J_i}\cdot~\mathbf{\hat n}~dA=\sum_i^N I_{\rm{out,i}}$$ The right hand side is the rate of change of the total charge enclosed within the volume. Thus, we write $$\sum_i^N I_{\rm{out,i}}=- \frac{dq_{\rm{encl.}}}{dt}.$$ That's the general form of KCL. We note that
1. When as much current goes into the volume as comes out, the derivative on the RHS is zero and ##\sum_i I_{\rm{out,i}}=0## with the understanding that charge going in the ##i##th location on the surface is added as ##-I_{\rm{out,i}}##. Of course in this case the other side of the equation is zero so the sign doesn't matter as long as charge out has opposite sign to charge in.
2. The sign
does matter when the enclosed charge is changing. Consider, for example, the case of volume ##V## enclosing only the positive plate of a
discharging capacitor through resistor ##R##. There is only one location where charge can come out, i.e. ##N=1##. Then ##\frac{dQ}{dt}## is a negative quantity which makes ##I_{\rm{out}}## positive. The current in the circuit must be drawn from the positive plate to the resistor before applying KVL.
I know I'm not saying anything new here, but it's nice to see that what might be considered "obvious" can be traced back to fundamental equations and principles that apply in general.
Edit: Minor typo fix.
