Re: Wire Length

["Tesla list" <tesla@xxxxxxxxxx>]

Original poster: Jared E Dwarshuis <jdwarshui@xxxxxxxxx>



The classic equation for an air cored inductor, derived with Maxwell's
equations is:

L = u Nsqrd Area / length

However the numerator and denominator can be multiplied by 4pi,
yielding:

L = u (2pi R N) sqrd / 4pi l

Since: 2pi R N is wire length ,  we can write:

L = u (wire length)sqrd / 4pi l*

I put a star next to the length because solenoids in the real world do
not have a perfectly uniform magnetic field. We then need to make our
solenoid length just a little bit longer to get the correct
inductance.

Now we can talk about standing wave resonance in a Tesla coil.  We
will use a simple version of capacitance in the lc equation.

We can use a sphere for our top end capacitor. The capacitance of a
sphere is:
 c = 4pi e R*

I put a star next to the radius because a Tesla coil inductor has self
capacitance that must be accounted for. We find that  R, in real life
is going to be a bit smaller due to the self capacitance of the
inductor.

Now we examine Tesla's equation:

C/4 Wire length = 1/ 2pi sqrt (lc)

Substituting from above for L and c, we get:

C/4 Wire = C/ Wire'  x  1/2pi  sqrt (l*/R*)

Set:  2pi = sqrt (l*/R*)

Then:

C/ 4 Wire = C/Wire'

Inverting frequency gives us the period:

4 Wire /C = Wire'/C

A casual inspection shows that this equation can only be satisfied if
Wire' = 4 Wire

Now we  apply the Lorentz equation, as we are observing time dilation
and distance contraction.

With a Gamma of 4, we predict that the velocity of waves down the
length of wire in our inductor will measure the sqrt(15/16)C

Thus, the actual wave velocity is reference frame dependant.

Tesla must not have understood all of this (in the late 1880's) or he
would have found the general form of the equation which describes node
formation.

n/2  C/w = 1/2pi sqrt ( u  x  (w/2n)sqrd x 2n/ 4pi l* x 4pi e  R*)

Where n = 1/2, 2/2, 3/3...........

The nodality equation also yeilds a gamma of 4


 Sincerely: Jared Dwarshuis (and by proxy),  Lawrence Morris