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Re: R = V/I ?



I'm somewhat late jumping into this thread, having been preoccupied with
a hard disk crash. My take on the issues is as follows:

In later courses a student is likely to come across the term "negative
differential resistance" in a context such as oscillator circuits. In
order for such a term to make any sense at all, it is necessary to allow
for R(V) or R(I). However, for the most part I see it as pointless to
talk about anything other than constant resistances and have to wonder
if the right things are being represented as fundamental when resistance
is made to depend on V or I.

By this time in the course you have probably covered force, energy,
power, charge, time, displacement, and a few other things - but these
are all fundamentals. Applied to a circuit we have definitions of
current (the amount of charge per unit time flowing through a circuit
element) and potential/voltage (the thing that dictates how much change
there will be in the electrical potential energy any given charge).
Everything is still fundamental - no special cases or approximations.
Clearly multiplying I*V will give us power
(charge/time * energy/charge = energy/time).

For a given circuit element we have a some characteristic cause-effect
relationship that will provide an I(V) or V(I) relationship. In general
this is not a straight line and may be history dependent (my own
measurements on light bulbs have found you are not guaranteed to
reproduce the same I-V curve if you measure 'I' while decreasing the
voltage from a hot bulb). However, in many cases the relationship is a
straight line through the origin. In such cases, for purposes of
utility, we refer to the proportionality constant as "resistance". By
measuring the proportionality constant we can describe the element's
behaviour with a single number rather than a table of values. The
proportionality constant is R = V/I - if this ratio is constant over
some range of V, then we would say that "Ohm's Law" applies to that
element over that range of voltages.

So the purpose of 'R' is purely utility - not that it is a fundamental.
If R is not constant, its utility disappears. What happens if you
measure R as a function of I in order to plot power versus I? You
measure voltage&current to calculate R, then use R to calculate power,
then plot power versus I... why not skip the middle step? Measure
voltage&current then multiply them to get power. You not only save work
but also emphasize the fundamentals by doing it this way. Knowing R as
a function of I or V doesn't add anything to knowing V as a function of
I, its just extra work which distracts from understanding whats going
on.

The other thing is the notion of how well students understand the
proportional to V and inversely proportional to I characteristic or
R=V/I. I would say that if students are even thinking in such terms at
all they have missed to boat conceptually. When it is useful to talk
about R, it is equal to V/I - it is not proportional or inversely
proportional to either of them individually.

For cases where Ohm's Law applies I like to spend time on the
microscopic basis. Going back to kinematics and making up a crude model
we get 'I' proportional to V(squared). So far from being fundamental,
Ohm's Law is contrary to what they should expect to observe based on
what they have learned in other parts of the course. However, add
collision effects leading to energy loss and you get a net drift speed
for the charge carriers which is proportional to the voltage - hence 'I'
proportional to 'V'. The net drift speed is a conceptual tie-back to
anything mentioned about terminal velocity for 'real' falling objects,
and this in turn provides opportunity to think about why drag effects
dominate in electronics, but we ignore them in trajectory problems...
(You may get the impression that I drift around a bit - I try to present
the material as a whole with many tie-ins, not as disconnected
individual topics.) Being a solid state physicist I also like to discuss
how this relates to the material property of "resistivity".

()-()-()-()-()-()-()-()-()-()-()-()-()-()-()-()

Doug Craigen
http://www.dctech.com/physics/