# Re: Variable Capacitance and Inductance

```Original poster: "Ed Phillips by way of Terry Fritz <twftesla-at-qwest-dot-net>" <evp-at-pacbell-dot-net>

Tesla list wrote:

> I need to interject a comment here.  Despite Matt's insinuation that anybody
> who takes a sincere interest in Tesla's work worships him as an infallible
> god, I have found several errors in Tesla's work in my studies.  Apparently
> I have found another one.  In reviewing Tesla's notes specifically related
> to measuring the capacitance of a "large structure" on page 282 of his CSN,
> he states, "The rise in the effective capacity for 47 feet and 6" was ...
> 26.2%.  Per one hundred feet it would be from this: 55.16% or a little over
> 1/2% per foot."  It seems Tesla incorrectly remembered the details of his
> notes and wrote in his article "50 per cent per foot of elevation" when he
> should have written "50 per cent per one hundred feet of elevation."
>
> His CSN correctly state the difference is 1/2% per foot, which is consistent
> with the rest of his article on variable capacitance.
>
> >He is now saying that the capacitance of the conductive body increases with
> elevation.

Indeed he does seem to be saying that he strung a 50 foot vertical wire
and moved a 30" ball up and down it, at each position adjusting the
inductance of the primary for resonance, and then estimating the total
capacitance of the ball and wire from the primary inductance required
for tuning.  I'm too tired right now to sketch out the full picture of
what he's saying, but one thing which is evident is that he is assuming
that the capacitance of his bottle primary capacitor is constant.
Perhaps someone who had played with bottle capacitors could comment on
their stability.  Note that he is NOT measuring the capacitance of the
wire plus ball (or other structures described in the following pages).
What he is doing is retuning the primary for "maximum good" and that
doesn't necessarily indicate that the capacitance was changing much.  He
makes no mention of the sharpness of tuning or the repeatability of his
settings.  Anyone who has played with a TC knows the tuning is not
necessarily sharp, so perhaps his apparent results were due to
experimental error.

> While checking his math I discovered Tesla was using the classic formula for
> inductance instead of a calibrated formula like Wheeler's (which wasn't
> discovered yet).

A couple of things.  First of all, the formula he used (page 253, for
example) is a very approximate heuristic one and correct ONLY if all of
the flux produced by the current flowing through one turn linked
completely to all of the other turns.  He must have been aware of that,
or at least he should have.  It is certainly far from correct for the
shapes of the coil he used.  The work by Lorentz, Rosa, and Nagaoka
among others was directed toward calculating the correct flux linkages
for any given geometry.  There are two common definitions of
self-inductance.  The most common one used when I was going to school
was that inductance was "flux linkages per unit current", which is
obviously affected by the geometry.  The second often used now is that
the inductance is the ratio of the voltage induced in a coil by a given
rate of change of the current through it.  Seems to me the first is more
fundamental.  As far as Wheeler's formulae (there are several), it IS
NOT based on empirical measurements, but on fitting simple expressions
to the more exact solutions over limited ranges of geometry.  The
smaller the range of L/D the more accurate the formula can be.  The
"normal form" is "good enough for government work" for coils whose
length is comparable to twice the diameter, and isn't bad for coils
longer and shorter than that.  He and others derived formulae for use on
short and long coils.

> Tesla apparently measured his inductances accurately as he
> noted his measurement was considerably less than the calculated inductance
> (page 253 CSN.)  Sorry Terry, but Tesla was aware that the EMF affected the
> inductance of the coil.  He makes such a comment on page 273 of CSN.  He
> probably was aware of this much sooner, so now I'll need to go back and look
> for other references.
>
> What's interesting is that Tesla made his correction for EMF from the
> classic inductance formula values and not from Wheeler's as you and Paul are
> doing now.  Why don't you try making your adjustment from the classic
> inductance formula of:
>
>     4 pi^2 N^2 R^2
> L = --------------
>           H
>
> Where L is in 1000 inches.

What does H stand for in this expression?  The length (height in his
case)?  As mentioned above, that is a very approximate formula used "for
example" and not for engineering calculation.  What do you mean by
"Where L is in 1000 inches".  Is L the inductance or the length or what?

> and see if the new value reveals some information?  This ties in with the
> variable inductance of a coil.  Not only is a coil affected by its own EMF,
> but it should also be affected by proximity to the earth.  It would appear
> that Wheeler's formula, being based on empirical data, is accounting for the
> coil's proximity to the earth.

See above.  Wheeler's formula is not empirical.  I've read through
these pages several times and I can find NO statement implying that the
thought the inductance was a function of the voltage across the coil.
His statement "Small corrections should have been made for the e.m.f.
making it smaller..."  seems to tie in with the statement at the bottom
of page 271 where he says "To ascertain approximately their inductances
readings were taken etc.".  In other words, he might be saying that the
inductance should be corrected by using the results of the e.m.f.
measurement (he was measuring the voltage due to a measured current
which would give the reactance from which the inductance can be
calculated).  He ends the previous paragraph with the statement that
"These readings do not quite agree with the result calculated, but I
think this only indicates some action of secondary on the primary when
the former is open, or else that the mutual induction coeff. measured a
little too high.  This is VERY LIKELY.)  Seems to me he is questioning
the accuracy of his measurements.
>
> This would imply that the closer to earth, the lower the inductance.  The
> further from the earth, the closer the coil gets to its unbiased inductance
> value.

I can't find the implication.

> When considering a static capacitor, one with no frequency applied to the
> charge, the closer the charged body is to the earth, the higher the
> capacity.  This is obvious.  The further the charged body is from the earth,
> the lower the capacitance will be.  However, Tesla was measuring a dynamic
> charged body.  The amount of charge a body will hold depends on the
> dielectric between the charged bodies.  The further the ball was raised, the
> more dielectric between the sphere and earth and hence the greater the
> capacity.

Not true.  The dielectric constant of the material surrounding the
conductor affects the capacitance for a given geometry.  The volume of
dielectric doesn't enter in per se.