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

H*rho := A*n*Pi*Diff(BesselJ(m,sqrt(omega^2/(c^2)-n...........(1a)

H*phi := -A*m*n*Pi*BesselJ(m,sqrt(omega^2/(c^2)-n^2.........(1b)

Hz := A*(omega^2/(c^2)-n^2*Pi^2/(h^2))*BesselJ(m,sq.........(1c)

E*rho := I*A*omega*uo*m*BesselJ(m,sqrt(omega^2/(c^2.........(1d)

E*phi := I*A*omega*uo*Diff(BesselJ(m,sqrt(omega^2/(......(1e)

Ez := 0...............(1f)

Cylindrical Stationary E Wave

E*rho := -A*n*Pi*Diff(BesselJ(m,sqrt(omega^2/(c^2)-..........(2a)

E*phi := A*m*n*Pi*BesselJ(m,sqrt(omega^2/(c^2)-n^2*..........(2b)

Ez := A*(omega^2/(c^2)-n^2*Pi^2/(h^2))*BesselJ(m,sq.........(2c)

H*rho := -I*A*omega*m*BesselJ(m,sqrt(omega^2/(c^2)-.........(2d)

H*phi := -I*A*omega*Diff(BesselJ(m,sqrt(omega^2/(c^........(2e)

Hz := 0................(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.

rho, phi, z -- The usual cylindrical coordinates ie rhois the radius of the cylindrical cavity, phiis 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.

omega-- The resonant angular frequency in radians per second ( 2*Pi*f)

2. The meaning of terms such as {sin(n*phi)/(OR-cos(n*phi))}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 phi, by pi/2radians (90 degrees) electrical. Because all field components are rotated together no new field configuration is normally produced. However, if m = 0 then sin(m phi) = 0 and cos(m phi) = 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 exp(I*omega*t)represents a rotating vector of magnitude one and rotating at a frequency omega. For the usual stationary fields of a standard cavity resonator it represents the sinusoidally varying amplitude with time. The H fields have a term I*exp(I*omega*t)and the Iterm represent a phase advance in time of the sinusoidal H field on the E field by Pi/2radians.

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 curlE = -diff(B,t)and curlB = diff(E,t)/(c^2).

5. The differential of the BesselJ expression is:-

Diff(BesselJ(m,sqrt(omega^2/(c^2)-n^2*Pi^2/(h^2))*r...= sqrt(omega^2/(c^2)-n^2*Pi^2/(h^2))*(-BesselJ(m+1,sq...

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 omega/mand differentiating with respect to phi. 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 Piradians (180 degrees) electrical with respect to the stationary fields.

2. The exp(I*omega*t)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 phi. omegawould 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 ( Omega) will be omega/mradians per second. This follows because there are m cycles of the field in each revolution in the phidirection. In fact it is because the field is sinusoidal in the phidirection 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 phidirection 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 omegawhich refers to the electrical variation we can, if preferred, replace all occurrences of it by Omegam which is still the electrical variation but expressed in terms of the mechanical spin frequency. Therefore exp(I*omega*t)becomes exp(I*m*Omega*t)etc.

3. The Iterm 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 Pi/2electrical in space relative to the E field. This is Pi/(2*m)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-

{cos(m*phi)/(OR*sin(m*phi))}*I*exp(I*omega*t)= {sin(m*phi)/(OR-cos(m*phi))}*exp(I*omega*t)

{sin(m*phi)/(OR-cos(m*phi))}*I*exp(I*omega*t)= -{cos(m*phi)/(OR*sin(m*phi))}*exp(I*omega*t)

4 The exponential terms such as sin(m*phi)*exp(I*m*Omega*t)represent rotation in the phidirection and can be written as the equivalent expression sin(m*phi-m*Omega*t). You can see this later expression also represents rotation as for t=0 say then at phi= 0 the field will be zero. However, as t increases phimust increase to keep sin(m*phi-m*Omega*t)equal to zero and the increasing value phiwith time is rotation.

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

Cylindrical Spinning H Wave

H*rho := -A*n*Pi*Diff(BesselJ(m,sqrt(m^2*Omega^2/(c...........(3a)

H*phi := A*m*n*Pi*BesselJ(m,sqrt(m^2*Omega^2/(c^2)-.........(3b)

Hz := -A*(m^2*Omega^2/(c^2)-n^2*Pi^2/(h^2))*BesselJ.........(3c)

E*rho := A*Omega*uo*m^2*BesselJ(m,sqrt(m^2*Omega^2/.........(3d)

E*phi := -A*m*Omega*uo*Diff(BesselJ(m,sqrt(m^2*Omeg......(3e)

Ez := 0...............(3f)

Cylindrical Spinning E Wave

E*rho := A*n*Pi*Diff(BesselJ(m,sqrt(m^2*Omega^2/(c^..........(4a)

E*phi := -A*m*n*Pi*BesselJ(m,sqrt(m^2*Omega^2/(c^2)..........(4b)

Ez := -A*(m^2*Omega^2/(c^2)-n^2*Pi^2/(h^2))*BesselJ.........(4c)

H*rho := -A*Omega*m^2*BesselJ(m,sqrt(m^2*Omega^2/(c.........(4d)

H*phi := A*m*Omega*Diff(BesselJ(m,sqrt(m^2*Omega^2/........(4e)

Hz := 0................(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.

Omega--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 m*phi-m*Omega*tterm. This has been done using a computer program written in Maple 6 and the equations comply with all four of Maxwell's equations.

 

 

 

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