Spherical Cavity Field Equations for Stationary and Spinning Modes

The field equations for the spherical cavity normal stationary resonant modes are well known and the general form of these equations is as follows:-

Conventional Stationary H Wave Equations

Er = 0...............(1a)

E*theta = -A*n*sqrt(c)*BesselJ(p+1/2,omega*r/c)*Leg................(1b)

E*phi = -A*sqrt(c)*BesselJ(p+1/2,omega*r/c)*Diff(Le..............(1c)

H*r = A*p*sqrt(c)*(p+1)*BesselJ(p+1/2,omega*r/c)*Le.............(1d)

H*theta = A*Diff(sqrt(r*c/omega)*BesselJ(p+1/2,omeg............(1e)

H*phi = -A*n*Diff(sqrt(r*c/omega)*BesselJ(p+1/2,ome............(1f)

 


Conventional Stationary E Wave Equations

Er := A*c^(3/2)*p*(p+1)*BesselJ(p+1/2,omega*r/c)*Le................(2a)

E*theta := A*c*Diff(sqrt(r*c/omega)*BesselJ(p+1/2,o.............(2b)

E*phi := -A*n*c*Diff(sqrt(r*c/omega)*BesselJ(p+1/2,.............(2c)

Hr = 0.............(2d)

H*theta := -A*n*BesselJ(p+1/2,omega*r/c)*LegendreP(..............(2e)

H*phi := -A*BesselJ(p+1/2,omega*r/c)*Diff(LegendreP..............(2f)

NOTES ON EQUATIONS:-

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

n, p  -- 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 controls the amplitude of all the field components.

r, φ, θ -- The usual spherical coordinates ie r is the radius of the spherical cavity, φ is the rotational angle about the z axis measured anticlockwise from the x axis, and θ is the angle to the z axis.

t     -- Time

ω    -- The resonant angular frequency in radians per second (2πf)

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 bottom terms are used instead of the top ones the only effect is that the bottom terms represent rotation of all the fields anti-clockwise, in the direction of  , by  radians electrical (i.e. 90 degrees electrical). Because all field components are rotated together no new field configuration is normally produced. However, if n = 0 then sin() = 0 and cos() =1 then looking at the main field equations it is apparent that 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. BesselJ functions are standard functions which often appear in solutions which use spherical coordinates. They define the form of the field in the radial direction. They are similar to a decaying sine wave and for example plots of BesselJ(-1/2,R), BesselJ(1/2,R), BesselJ((3/2,R) and BesselJ(5/2,R) are shown below in Fig 20.

 


Note that most of the BesselJ terms in the field equations are also divided by r, , or  r^(3/2)which will alter the radial rate of decay of the field distribution from that of the BesselJ function alone.

4.  LegendreP(p,n,cos(theta))are known as associated Legendre Polynomials and they define the field distribution in the θ direction. They are more frequently written in text books as Ppn(cos( θ)). The LegendreP functions are again standard functions and are just polynomials in cos(θ) and sin(θ) and their value may be looked up in tables. They only have valid real values for n  p and typical functions are:-

LegendreP(p,n,cos(theta))

Function

p

n

0

0

1

1

0

cos(θ)

1

1

-sin(θ)

2

0

(3cos2θ+1)

2

1

-sin2θ

2

2

3

0

3

1

3

2

15

3

3

5. The term exp(I*omega*t)represents a rotating vector of magnitude one and rotating at a frequency ω. For the usual stationary fields of a standard cavity resonator it represents a 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  radians.

6. 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).

7. It is usually easiest to find the sine and cosine equivalent of the Legendre functions directly from tables and then take the differential if required. It may be of interest to note that the differential of the Legendre function is:-

Diff(LegendreP(p,n,cos(theta)),theta) = ((p-n+1)*Le...

The differential of the BesselJ expression which is often used is:-

Diff(sqrt(r*c/omega)*BesselJ(p+1/2,omega*r/c),r)= (-c*p*BesselJ(p+1/2,omega*r/c)+omega*r*BesselJ(p-1/...

If using a Maple computer program it is useful to remember that:-

            sin(nv - nωt) = -sin(-nv + nωt)

but       cos(nv - nωt) = cos(-nv + nωt)

Spinning Resonant Spherical Fields

The above equations and notes represent the standard interpretation of the fields in a resonant cavity. However, the same equations can be used to describe a spinning field. In this case the exp(I*omega*t)term 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 ф. Now ω is 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/nradians per second. This follows because there are n cycles of the field in each revolution in the ф direction. In fact it is because the field is sinusoidal in the ф direction that rotation in this direction is to the observer the same as a field variation in time. If n = 0 then there is no variation of the field in the ф direction so no field variation with rotation and therefore this is not a possible spinning mode. All higher values of n i.e. 1,2,3,4 etc will support spinning fields. For the spinning field instead of using ω which refers to the electrical variation we can, if preferred, replace it by  nOmegawhich is still the electrical variation but expressed in terms of the mechanical spin frequency. Therefore exp(I*omega*t)becomes exp(I*n*Omega*t) etc.

The Iterm in the H field equations (1d to 1f) and (2e and 2f) 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 in advance of the E field. This is Pi/(2*n)radians of mechanical rotation.

The above description is the essentials of the spinning field description but we can carry out this transformation from stationary to spinning field in a more mathematical manner and at the same time it is possible to express the field equations in a slightly different way which emphasizes the rotating aspects.

For any field which has a varying amplitude distribution through space and which moves in a straight line relative to the observer then a change in the field with time is given by:-

 =  Diff(E,t)+Diff(E,x)*Diff(x,t) = Diff(E,t)+v*Diff(E,...  

For a field just rotating in the v direction the total change is:-

  =  Diff(E,t)+Diff(E,phi)*Diff(phi,t) = Diff(E,t)+Omega...  

or, as discussed above, as the mechanical speed of rotation, Omega= omega/nthen the rate of change of E may be written:-

    =   +   ..........(3)

The rate of change is therefore in two parts. The first part, , is just due to changes with time and the second part    is due to changes in location. The normal stationary fields just involve the first term and the spinning fields just involve the second term.

In applying equation (3) it is initially only necessary to consider the parts of the field that are functions of ф or t. The stationary E and H fields of the H wave equation (1) above can therefore be represented by simplified expressions. The radial and theta parts of each equation has been replaced by functions F r, Fθ and Fф and F'r, F'θ and F'ф and these can be considered constant for the mathematical operations about to be carried out. So the simplified fields are:-

E =      0,                  -Fθ {sin(n*phi)/(OR-cos(n*phi))}exp(I*omega*t), -Fф {cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t)...........(4a)

H = F'r {cos(n*phi)/(OR*sin(n*phi))}*I*exp(I*omega*t),F' θ{cos(n*phi)/(OR*sin(n*phi))}*I*exp(I*omega*t),-F'ф {sin(n*phi)/(OR-cos(n*phi))}*I*exp(I*omega*t).........(4b)

Differentiating the E and H fields in equation (4a) and (4b) with respect to t, to find the change with time for the first part of eqn(3), which occurs in the normal  stationary cavity :-

  Diff(E,t)=        0,              -Fθ I*omega{sin(n*phi)/(OR-cos(n*phi))}exp(I*omega*t), -Fф I*omega{cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t)..............(5a)

  Diff(H,t)= - F' r omega{cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t),-F'θ omega{cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t)          ,+F'ф omega{sin(n*phi)/(OR-cos(n*phi))}*exp(I*omega*t).........(5b)

Differentiating equation (4a) and (4b) with respect to ф and multiplying by   to find the second part of eqn(3), which is the change due to the constant amplitude spinning fields :-

    [omega/n]*Diff(E,phi) =       0,          -Fθ omega{cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t), +Fф omega{sin(n*phi)/(OR-cos(n*phi))}exp(I*omega*t).......(6a)

    [omega/n]*Diff(H,phi)= - F' r omega{sin(n*phi)/(OR-cos(n*phi))}*I*exp(I*omega*t),-F'θ omega{sin(n*phi)/(OR-cos(n*phi))}*I*exp(I*omega*t),-F'ф omega{cos(n*phi)/(OR*sin(n*phi))}*I*exp(I*omega*t).........(6b)

The following relationships can be substituted in eqns (6a) and (6b), where I is rotation by  radians in the ф direction:-

            sin(nv) =  I cos(nv)  …….(7a)

            cos(nv) = -I sin(nv)  ....….(7b)

NB. Care is necessary when using this substitution as although it is correct for rotation, as in this case, it is not numerically correct (i.e. sin(n*phi)is not numerically equal to sqrt(-1)*cos(n*phi)although cos() phase shifted or rotated to advance by  electrical is the same as sin() ). It is also not valid for n = 0 as phase advancing a cos() = 1 field which will have a constant amplitude of one by  electrical cannot give a sin() = 0  field of constant amplitude zero. However, as a spinning field does not exist for n = 0 this limitation does not affect the results.

This substitution gives:-

   [omega/n]*Diff(E,phi) =        0 ,         +Fθ omegaI{sin(n*phi)/(OR-cos(n*phi))}*exp(I*omega*t), +Fф I*omega{cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t)......(8a)

   [omega/n]*Diff(H,phi)= + F' r omega{cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t),+F'θ omega{cos(n*phi)/(OR*sin(n*phi))}*exp(I*omega*t),-F'ф omega{sin(n*phi)/(OR-cos(n*phi))}*exp(I*omega*t).........(8b)

A sufficient condition for two fields to be the same is that the divergence and curl of the fields are the same. For the stationary and spinning fields the divergence of E and H will be the same, i.e. zero, as required for charge free fields. From Maxwell’s equations we know     and . Therefore, if  for the spinning and stationary fields are the same then curlH is the same and hence the spinning and stationary H fields are the same. Similarly the E fields are identical if   is the same.

Comparing equations (8a) with (5a) and (8b) with (5b) we see that the time rates of change are identical apart from all terms in equations (8a) and (8b) being of opposite sign. This shows that the spinning fields are of the identical form to the conventional stationary cavity fields. The minus sign just indicates that the spinning E and H fields are both rotated by π electrical (180 degrees) in space with respect to the stationary fields

Carrying out a similar calculation for the alternative bottom ф term in the H wave equations (1) and both top and bottom E wave equations (2) shows that again the spherical cavity spinning fields are of identical form to the stationary fields apart from the minus sign.

Writing Spinning Field Equations.

Although it has now been shown that the conventional cavity field equations (1) are applicable to a rotating field there are a number of changes which can be made to the equations to emphasize the spinning aspect:-

1. The H field expressions contain an I term which indicates that the spinning H field is  degrees electrical in advance of the E field. This phase advance is essential for the fields to spin but can be expressed in a different way by removing the I term using equations (7a) and (7b) so:-

 I e (Iωt)        =     e (Iωt)

 I e (Iωt)    = -  e (Iωt)     

2. The mechanical speed of rotation, Omega= omega/n, so replace all occurrences of ω in the equations by n*Omega. The equations are then written in terms of the angular frequency of revolution.

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

Making these three changes to eqn (1) and changing the sign we have the field equations for a spinning field:-

Spinning H Wave Equations for a Spherical Cavity

Er = 0...............(9a)

E*theta = A*n*sqrt(c)*BesselJ(p+1/2,n*Omega*r/c)*Le................(9b)

E*phi = A*sqrt(c)*BesselJ(p+1/2,n*Omega*r/c)*Diff(L..............(9c)

H*r = -A*p*sqrt(c)*(p+1)*BesselJ(p+1/2,n*Omega*r/c).............(9d)

H*theta = -A*Diff(sqrt(r*c/(n*Omega))*BesselJ(p+1/2........(9e)

H*phi = -A*n*Diff(sqrt(r*c/(n*Omega))*BesselJ(p+1/2............(9f)


Spinning E Wave Equations for a Spherical Cavity

Er := -A*c^(3/2)*p*(p+1)*BesselJ(p+1/2,n*Omega*r/c)................(10a)

E*theta := -A*c*Diff(sqrt(r*c/(n*Omega))*BesselJ(p+.......(10b)

E*phi := A*n*c*Diff(sqrt(r*c/(n*Omega))*BesselJ(p+1............(10c)

Hr = 0.............(10d)

H*theta := -A*n*BesselJ(p+1/2,n*Omega*r/c)*Legendre..............(10e)

H*phi := A*BesselJ(p+1/2,n*Omega*r/c)*Diff(Legendre..............(10f)

 

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 (9) and (10) satisfy Maxwell's equations. For this it is, of course, necessary to use the full equations with either the top or bottom n*phi-n*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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