I hope I do not sound too snarky, and none of this has ever helped me fix a car, but…

George Simon Ohm observed that for many materials the density of the electrical current, J, (a vector quantity) was directly proportional to the applied electric field, E, (a vector quantity). The (scalar) proportionality constant is called the conductivity, sigma. The original “point” form of Ohm’s Law is J=sigma*E. When considering total currents, etc, this becomes the more familiar V=I*R.

That linear relationship is really handy. It makes lots of problems mathematically tractable; you can solve them. The materials for which the linear “Law” holds are called Ohmic conductors – a rather circular definition.

But not all conductors are Ohmic. For non-linear conductors (e.g., a solid-state diode at low-ish forward current), V and I are not proportional; an increase of, say, two-fold, in V does not give a two-fold increase in I. However, because we like to use the linear Ohm’s “Law”, we use the slope of the V-I curve at any point as the “incremental” resistance, and say that Ohm’s Law applies to small-enough CHANGES in V or I around that point. “Small enough” depends on how good an approximation you need.

Similarly, we like to use a linear model for the voltage drop at a battery’s terminals as the load current increases. So, we define an “internal resistance”, but, again, it is not the same for all values of current. And I suspect (but do not know) that for our lead-acid automobile storage batteries the internal resistance depends also on the battery’s state of charge and also on a bunch of things related to its age and use history.

For those more familiar with physical chemistry, we do the same kind of linear approximations with the heat constant, c, as in delta-Q = m*c*delta-T). Look in the tables, and you will find that the value of c for water depends on the temperature. We are just making the linear approximation so we can solve the problem! (I had this epiphany about 20 years ago while preparing to tutor a high school chemistry student.)

Ohm’s Law applies in many situations; properly understood and applied, it is very useful. But it will not tell us directly the relationship between the battery’s terminal voltage and the current in/out of the battery. Which, I believe, is where I came in.