I am going to try to answer Jim Green's questions, and try to summarize
where I am with respect to this thread.
(1) One way to describe a device that follows Ohm's Law is to say its plot
of I-versus-V must be a straight line.
A light bulb clearly does not have a linear plot. But neither does anything
else unless we hold its temperature constant. I think this is a source of
confusion because it divides us into two camps. One camp seems content to
ignore the nonlinearity of I-versus-V for devices that have small
temperature changes. They say the light bulb does not follow Ohm's Law, but
a standard electronics resistor does. The other camp is inclined classify
light bulbs and normal resistors as equivalent once temperature is
considered. Neither have truly linear plots, although the standard
electronics resistor can be outfitted with a heatsink so its temperature
varies only slightly, and the nonlinearity of I-versus-V is not detected
within the accuracy of standard digital multimeters.
My reason for being in the second camp should get more clear in the
following points.
(2) Another way of describing Ohm's Law (the one I prefer) is to say the
resistance must be independent of the electric field (E).
This is equivalent to saying the resistivity must be independent of the
electric field, because R = rho*L/A, and L and A are clearly field
independent.
In the classical picture we can derive rho = m*v(ave)/(n*e^2*lambda) where m
is the mass of the electron, v is the mean speed of the electron, n is the
number of electrons per unit volume, e is the charge of the electron, and
lambda is the mean free path of the electron. For a discussion of this,
see, for example, Tipler, Fourth Edition pages 830-832.
Saying rho is independent of the field then becomes equivalent to saying the
mean speed and the mean free path must be independent of the electric field.
This would be the case for the typical conductor. The main problem with
this picture is it gives the wrong temperature dependence when we try to
combine it with Maxwell-Boltzman velocity-versus-temperature calculations.
Although this is a big problem, the general idea that the resistance is
caused by electrons weaving their way through the lattice of the conductor
seems reasonably sound to me, and it is the way I view the idea of "pure
resistance." And, of course, the picture can be made more correct/accurate
by considering electrons as waves, etc.
In this picture, the tungsten of a light bulb, and the nichrome of a
resistor, and a piece of copper wire are all behaving essentially the same.
If we increase the potential difference and if we note the resistivity
changes, it is not because we increased the electric field per se, it is
because we had an increase in temperature. Repeat, the increased electric
field does not directly cause the increase in resistivity; rather the
increased temperature causes the increase in resistivity. If we are able to
hold the temperature constant, the resistivity will not change and a linear
I-versus-V plot will result.
In this view, a light bulb follows Ohms Law because its resistance does not
change as a result of changes in the electric field within the conductor.
Its resistance changes because of temperature.
(3) There are devices for which the apparent resistance does change because
of the electric field. A zener diode is a simple example. The picture of
conduction in this case is quite different because now another aspect of the
rho equation (above) changes... the number of charge carriers per unit
volume, n. With the electric fields present at the junction when we are
below the zener voltage, there are few charge carriers. As we raise the
voltage through the zener voltage, the number of charge carriers increases
dramatically. Devices like this do not follow the model described in
section (2) in which we assumed a fixed number of charge carriers per unit
volume weaving their way through the lattice.
In these cases I do not calculate, measure, or assign a resistance to the
device. I thought I was amongst large company in this practice, but this
thread has made me wonder if that is true. Although we could conceivably
determine the parameters in the rho equation, and we could calculate a value
of R from rho*L/A, and we could also calculate a value for R from V/I, in
the circles of physicists I've been associated with, we just don't do that.
Once we get into a device in which the charge carriers per unit volume is a
function of the electric field, the classical idea of resistance is out the
window, and Ohm's Law, which is a classical expression, is also out the
window.
Therefore, in my mind, and I still think quite a few others, we only assign
a resistance to conductors/devices for which the classical model of
conduction is a reasonable model. Perhaps that is the best way to say it.
I also tried to word it another way by saying we ought to look at why a
potential difference exists across a device. Is it because electrons are
weaving their way through the lattice? If so, it is a classical resistor
and we can assign a resistance as R = V/I. However, if the potential
difference is caused an electrochemical reaction (battery) or lack of charge
carriers (diodes, transistors), etc. then this is not a classical resistor
and assigning a resistance as R = V/I is not appropriate.
(4) Jim Green is correct that power dissipation can occur in many devices,
including ones that I do not define as following Ohms Law. I think the
whole idea of power is a completely separate issue. The only reason it
might appear to be related is because of the tendency of many people to
think primarily of I^2*R when they think of power. In my way of thinking,
I^2*R is only appropriate for classical resistors. The generic formula for
power is I*V.
I think this is more than semantics. delta-V is (delta-U/q). I is
q/(delta-t). I*V is (delta-U)/(delta-t) = power. This will be true for any
device with a current through it and a potential difference across it. The
unit of volt is joules-per-coulomb. When a certain number of coulombs
experience a potential change (voltage change) there is an energy change.
If this happens in a specific time period, we have a defined power.
However, substituting I*R for delta-V in order to turn I*V into I^2*R is
only appropriate for devices for which it is appropriate to use the concept
of R. For example, the power dissipated in a true resistor is I*V which can
also be calculated as I^2*R. But, for a zener diode the power is still I*V,
but it is not I^2*R because the zener diode does not have a resistance.
Likewise the power supplied by a battery that is providing energy, or the
power received by a battery that is being recharged, is I*V. However, the
power supplied/received by a battery is not I^2*R because a battery does not
have a resistance defined by V/I.
A battery is also interesting because it typically heats up when it is being
charged or discharged. That's because a battery does have an internal
resistance in the classical sense. But this internal resistance is
definitely not V/I. Hence, when we supply power to recharge a battery, some
of the power goes into the chemical reaction, and some of the power goes
into thermal energy. The calculation I*V is correct for the total power
provided to the battery. If we know the emf of the electrochemical cell, we
can calculate the amount of power that is going into the reaction as I*emf,
and then the rest goes into thermal energy. We can calculate the internal
resistance of the battery by assuming the difference between I*V and I*emf
is I^2*R with R being the internal resistance (a classical resistance).
I think this battery example is a good one because it ought to make it clear
there is a difference between I*V and I^2*R.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817