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Re: Light bulb ohmicity (very long)

David Bowman has commented on my posting below, and I have left
intact the relevant parts, though this gets a bit lengthy. It
appears I did not explain my points well, so I will do so here
at more length. Contrary to the opinion of the originator of
the parent thread (note I changed the subject line) I do think
this is an interesting topic, and as well a useful one to be
discussed by teachers, especially if it can attract comments
from a person of David's great acumen.

I wrote, in differing from David's interpretation of the claim
that light bulbs do not display ohmic behavior:

It is perilous to do so, but in the interest of removing what
I think may be a nascent misconception, I will have to disagree
with David Bowman here. Tungsten may be an "ohmic substance"
(though I am unable to define what that means). I'll say more
about it later later. Light bulbs, even though their filaments
are made of tungsten, are *not* ohmic devices.

to which David replied

Since you just admitted that you don't know what it means to be 'ohmic',
how can you be sure that tungsten filaments are not ohmic? BTW, what is
the "nascent misconception" that you had in mind?

Actually the term "ohmic" is used in experimental physics to mean
linearity of the I-V characteristic. When I say I make "ohmic
contact" to something I mean that the contacts introduce no
appreciable nonlinearity of their own. I've used the term "ohmic"
in print in just that way. I don't know what it means in the
context in which I think you are using it, the materials context.

Ohm's law is not
a law of nature any more than Hooke's law is. Both are simply
useful approximations of actual behaviour, and the degree to
which a particular resistor (or spring) is ohmic (or Hookeish)
has reasonable tolerances.

Very true.

Light bulbs operate in a range of
currents for which their behaviour would most certainly be
considered non-ohmic by any engineer,

I'm not all that surprised.

and by this physicist as
well.

OK, I'm surprised.


A measure of "ohmicity" might well be in order here. I'm at
home today nursing a cold, and I don't have access to the
references (and laboratory) I would like, but a good measure
might be the quantity

V dI
r' = --- ----
I dV

I like your definition as long as an isothermality condition is included.
I seems though that maybe r' - 1 is more properly a measure of
*non*ohmicity. Maybe 1/|r' - 1| would be a good candidate for the
ohmicity.

I did consider the alternatives you proposed, but since I believe
an isothermality condition to be incompatible with physically
realistic use, I did not include that.

I think the difference between our views lies in the use of the
term "Ohm's law" for two different but closely related approximate
relations used in physics*. One, the more common of the two,
relates to electrical components. It is the one to which I refer
implicitly above:

V = I R

In this company I need not say more about that one. The second,
called Ohm's law by solid state theorists, is usually written:
-> ->
j = sigma E
-> ->
where j is the current density in a conductor, E the electric
field intensity, and sigma the conductivity of the conductor
(sigma is just the reciprocal of the resistivity, of course).
The conductivity is temperature dependent, and to the extent
that it is current density independent the conductor *may* be
said to be ohmic, though in my experience solid state theorists
usually do not use that term. The ohmicity r" under those
circumstances *might* be defined

E dj
r" = --- ---- .
j dE

Of course Ohm's law in anisotropic materials requires the use
of a conductivity matrix, but we won't explore that avenue on
this excursion.

My comments on "ohmicity" refer entirely to the first use of
Ohm's law. It is only appropriate to consider this on the
device level, not the materials level, as I hope to convince
you below.

. . . . Light bulbs in the vicinity of their operating
points deviate greatly from the ideal, though without a
laboratory I can't estimate how greatly.

I disagree here. Since I want the isothermality condition enforced when
the ohmicity of the bulb is measured I suggest that the bulb be operated
with an AC power source whose frequency is high enough so that the
filament's temperature does not fluctuate appreciably between the peaks
and the zero-crossings of the AC signal. I suspect that a 120 Hz thermal
cycling rate (twice the 60 Hz North American line freq.) may be somewhat
too low for an accurate ohmicity measure. The frequency ought not be so
high, though, that reactive effects start showing up due to the nonzero
inductance and capacitance possessed by the coiled geometry of most bulb
filaments. My guess is that a frequency of the order of 1 kHz would be
sufficient. To measure the ohmicity one could simply sample the current
through the filament simultaneously with the instantaneous voltage across
it at all phases of the AC signal. The slope of a log-log plot of |I(t)|
vs |V(t)| would then display ohmicity r'. Note, I predict that if the
current signal and the voltage signal are each fed to a separate channel of
a two channel oscope and the resulting traces are observed on the screen,
a judicious scaling and shifting of the two traces could superimpose both
of them making the screen look like a single trace. This would be an
explicit demonstration of ohmicity. If the tungsten is truly nonohmic (or
if the scope has some systematic nonlinearities) then the nonlinearity in
the I- V function would make both signals have a different shape and thus
they could not be adequately superimposed. If the frequency is wrong for
the job there would be a phase shift between the traces of each channel due
to reactive effects if the frequency is too high, or possibly due to a time
delay between the peak surface temperature and the peak Joule heating power
due to some thermal "inertia" if the frequency is too low and the
temperature is not held very constant throughout the heating cycle. For
this experiment a sine wave voltage would not result in a sine wave current
if either the tungsten or the scope were truly nonlinear.

You have not considered hysteresis (by which I do not mean the
specific magnetic use of that term) in the discussion above, but
I do compliment you on your evident appreciation** of the not
inconsiderable care which must be taken by the experimentalist in
exploring this phenomenon. However I believe you have now imposed
so many ideal constraints on this measurement that it can no
longer be said to be pertinent to or representative of the light
bulb in question. The contention is that light bulbs do not obey
Ohm's law; it is that position I wish to defend.

First, the ordinary light bulb runs at 60 Hz. This is so far
short of being fast enough to reach thermal steady state, as you
guessed, that I have seen its effects in high shutter speed
videotapes. The illumination brightens and dims slowly as the
field frequency beats with line frequency. (This was done with a
halogen lamp illuminant. The effect is greater for such a bulb
than it would be for a non halogen bulb.)

Hysteresis is important here because of heating. The temperature
of the filament's surface will vary with a phase lag with
respect to that of the current squared, the reason being that the
thermal conductivity of the filament requires that there be a
higher temperature at the filament's center than at its surface.

David's idea of immersing a tungsten filament in a constant
temperature environment is invincibly problematic. It would be
possible to do such a thing if Joule heating could be ignored,
but it can't. I appreciate the theoretical approach to many
problems, but a dissipationless flow of current in an ohmic
substance is, I'm afraid, out there with the spherical cow.

I don't care so much about whether or not the environment is the same
constant temperature at the filament. I really only care that the
filament be held at a constant temperature. My suggestion for
accomplishing this is to take the data for the experiment fast enough
so that the temperature doesn't have time to change. Hence the AC
measurement method describe above.

Since heat is produced at a not inconsiderable rate (power
density = j E = j^2 / sigma) and there is no phase change taking
place, thermal equilibrium would require the magical removal of
energy, since finite thermal conductivity limits the rate at
which it can be removed by other means. In the AC case the
specific heat of the metal also matters.

In consideration of this energy disposal problem, do you still
feel that a requirement of isothermality serves any physically
useful end in discussion of the ohmicity of a material? Do you
feel that some materials would be found to be ohmic and some
not? If your answer to that question is "Yes", please give me
some examples (I can only think of one). If your answer is "No"
(or if my example is the only one extant) then I would say the
concept is not useful.

The nascent misconception I perceived and set out to remedy has
now been thoroughly discussed, but I have not lost interest in
it. In fact I assigned my students the task of finding the
relation among the quantities involved in the case of direct
current flow in a cylindrical light bulb filament. The practical
case is more difficult, since as David noted a filament is coiled
(actually it is doubly coiled, as examination with a magnifier
will reveal). The physics I've given my students so far only
tretas radiative energy transfer from convex bodies, so the next
problem I'll give them is to try to understand the real filament.

Far from being boring, I think there is lots of, um, enlightening
discussion which can emerge from contemplation of the engineering
marvel that has evolved since its invention by the Wizard of
Menlo Park.

Leigh

* There is a third Ohm's law which is found in psychoacoustics,
but save that for another discussion.

** For the casual reader's edification I will say that it is my
impression that David Bowman is a theoretician. I (Leigh) am an
experimentalist. In some departments these are considered to be
two different religions, and this discussion, as important as the
Swiftian conflict between the big endians and the little endians
which led to bloodshed, is exemplary of that friction.