__Cylindrical
Cavity Field Equations for Stationary and Spinning Modes__

This analysis
of the cylindrical cavity is based on that for the spherical cavity. The
procedure used is very similar to that in 'Spherical Cavity Field
Equations' which should be consulted for details
of the analysis as this page mainly contains just the results. The general form
of the well known field equations for a cylindrical cavity normal stationary
resonant modes are as follows:-

__Cylindrical
Stationary H Wave__

........(1a)

......(1b)

......(1c)

......(1d)

...(1e)

...............(1f)

__Cylindrical
Stationary E Wave__

.......(2a)

.......(2b)

......(2c)

......(2d)

.....(2e)

................(2f)

__NOTES ON
EQUATIONS:-__

1. The symbols
used in the equations are as follows:-

m,n -- Can take
the value 0, 1, 2, 3.....etc and are the integers defining the harmonic
solutions. (Sometimes called the eigenvalues)

c -- The
velocity of light

A -- An
arbitrary constant which determines the amplitude of all the field components.

, z -- The usual cylindrical coordinates
ie is
the radius of the cylindrical cavity, is the rotational angle about the z axis measured anticlockwise
from the x axis, and z is the usual z axis along which the cylinder height is
measured.

t -- Time

h --The height
of the cylindrical field.

-- The resonant angular frequency in radians per second (
)

2. The meaning
of terms such as is
that either the top term is used for each field component or the bottom term is
used. For a particular resonant mode it is not permissible to mix terms from
the top and bottom for the E and H fields. The minus sign included with the
bottom term indicates that this field component is multiplied by -1. Although
the equations look different if the top terms are used instead of the bottom
the only effect is that the bottom terms represent rotation of all the fields
clockwise, in the direction of , by radians (90 degrees) electrical. Because all field
components are rotated together no new field configuration is normally
produced. However, if m = 0 then sin(m ) = 0 and cos(m ) = 1 so the top terms produce a valid field whereas the
bottom terms do not. It is therefore generally preferable to use the top terms
rather than the bottom. .

3. The term represents a rotating vector of
magnitude one and rotating at a frequency . For the usual stationary fields of a standard cavity
resonator it represents the sinusoidally varying amplitude with time. The H
fields have a term and the term represent a phase advance in time of the sinusoidal H
field on the E field by radians.

4. The above
equations define a field which has zero divergence of both electric and
magnetic fields and satisfy Maxwell’s equations for a field in vacuum,
without charges or currents, of and .

5. The
differential of the BesselJ expression is:-

=

__Re-writting The Spinning Field Equations.__

The above
equations and notes represent the standard interpretation of the fields in a
resonant cylindrical cavity. However, in the same way as was shown for the
spherical cavity the cylindrical cavity equations can also describe a spinning
field .The changes which can be made to the equations to more readily be seen
to represent a spinning field are as follows:-

1. We can
compare the rate of change of the above fields with time by differentiating
with respect to t and comparing this with the rate of change due to spin by
multiplying by and differentiating with
respect to . We
again find that the two rates of change are identical and this therefore means
that the spinning and stationary cylindrical fields themselves have identical
field distributions. We do still get a minus sign for the spinning fields rate
of change showing that the spinning fields are just rotated radians (180 degrees) electrical
with respect to the stationary fields.

2. The term now refers to the field change that
occurs due to the field spinning in space with constant amplitude about the z
axis in the direction of . would
be the frequency of the field as sensed by a probe positioned so as to measure
the frequency of the field rotating past it. This can be considered as the
electrical speed of rotation and the mechanical frequency of spin ( ) will be radians per second. This
follows because there are m cycles of the field in each revolution in the direction. In fact it is because
the field is sinusoidal in the direction that rotation in this direction is to the
observer the same as a field variation in time. If m = 0 then there is no
variation of the field in the direction so no field variation with rotation and therefore
this is not a possible spinning mode. All higher values of m ie 1,2,3,4 etc
will produce spinning fields. For the spinning field instead of using which refers to the electrical
variation we can, if preferred, replace all occurrences of it by m which is still the
electrical variation but expressed in terms of the mechanical spin frequency.
Therefore becomes etc.

3. The term in the E field equations 1d and 1e and the H field
equations 2d and 2e above referred to a time phase difference between fields.
It now refers to a space phase difference which requires that the H field is
rotated electrical
in space relative to the E field. This is radians of mechanical rotation. This
phase difference is essential for the fields to spin but can be expressed in a
different way by removing the I term using the following equations which apply
to rotating fields unless m = 0. As already stated m = 0 is not a valid
spinning mode so the substitution is valid-

=

=

4 The
exponential terms such as represent rotation in the direction and can be written as the equivalent expression . You can see
this later expression also represents rotation as for t=0 say then at = 0 the field will be zero.
However, as t increases must increase to keep equal to zero
and the increasing value with time is rotation.

Making these
four changes to eqn (1) we have the following field equations for a spinning
field:-

__Cylindrical
Spinning H Wave__

........(3a)

......(3b)

......(3c)

......(3d)

...(3e)

...............(3f)

__Cylindrical
Spinning E Wave__

.......(4a)

.......(4b)

......(4c)

......(4d)

.....(4e)

................(4f)

__NOTES ON
EQUATIONS:-__

m,n -- m can
take the value 1, 2, 3.....etc and n can take the values 0, 1, 2, 3....etc.

--the mechanical rotational angular frequency.

In order to
confirm that the field equations have been correctly manipulated it is only
necessary to check that the E and H fields in equations (3) and (4) satisfy
Maxwell's equations. For this it is necessary to use the full equations with
either the top or bottom term. This has
been done using a computer program written in Maple 6 and the equations comply
with all four of Maxwell's equations.