Re: Quarter Wavelength Frequency

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk> 

Gerry wrote:

 > There is only one fundamental resonance.

Yes, the currents and the fields are locked together in a
manner described by the Maxwell/Lorentz equations.  Given
the current and charge distributions you can calculate the
field, and vica versa, so there is no room to independently
vary one without the other following suit.

The helical conductor imposes boundary conditions on the field
which lead to a spectrum of resonances in much the same way
as the walls of a chamber impose boundary conditions on the air
within to produce a set of acoustic resonances.

Each resonance can be described in terms of the motion of the
field or the motion of the charges - the two descriptions are
equivalent and interchangeable.  From the point of view of
the charges, the presence and motion of charge in one part of
the winding affects the motion of charges elsewhere in the
conductor, and the set of 'mechanical' constraints thus imposed
forces the system to obey a differential equation in the charges
and currents.   Alternatively, we can set up a differential
equation in the field variables by combining Maxwells equations
with the constraints imposed by the wire.  The two equations
lead to the same solutions because they describe the same
set of resonances in equivalent terms.

We can (when certain conditions are satisfied) express the
remote coupling between charges through the 'mechanism' of
the field by means of the abstractions of capacitance and
inductance - concepts which neatly encapsulate all the relevant
Maxwellian detail of the remote interactions between the charges
along the wires.  This saves us a great deal of work - we can
operate with 1-dimensional arrays of 2-component quantities
(current and charge) rather than 3D volumes or 2D surfaces
of 3-component field vectors.

But we mustn't forget that these 'circuit theory' models are
no more than a way to represent the behaviour of the EM field
in terms of the behaviour of the associated charge movements.
If we forget this, we might find ourselves suggesting that the
currents can do things independently of the fields.

When we come to apply circuit theory to TCs, we calculate
the mutual capacitance between any two points x and y on
the coil.  In fact we do that for *every* possible pair of
points x and y.   Then we do a similar thing to get the mutual
inductance between every pair of points on the coil.  These two
'mutual reactance distribution functions' tell us all we need
to know about how the charges throughout the coil affect one
another.   They go into a straightforward but tedious calculation
out of which pops all the resonances, charge and current
distributions, impedances, and so on.  Then if we want to know
what the the field is doing, we just calculate that from the
currents and charges.
--
Paul Nicholson
--