UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 10.5 Prof. Steven Errede LECTURE NOTES 10.5

LECTURE NOTES 10.5

EM Standing Waves in Resonant Cavities

 One can create a resonantcavity for EM waves by taking a waveguide (of arbitrary shape) and closing/capping off the two open ends of the waveguide.

 StandingEM waves exist in (excited) resonant cavity (= linear superposition of two counter-propagating traveling EM waves of same frequency).

 Analogous to standing acoustical/sound waves in an acoustical enclosure.

 Rectangular resonant cavity – use Cartesian coordinates

 Cylindrical resonant cavity – use cylindrical coordinates to solve the EM wave eqn.  Spherical resonant cavity – use spherical coordinates

A.) Rectangular Resonant Cavity: (L W H   a b d) with perfectly conducting walls

(i.e. no dissipation/energy loss mechanisms present), with 0 x a, 0 y b, 0 z d.

n.b. Again, by convention: a > b > d.

Since we have rectangular symmetry, we use Cartesian coordinates - seek monochromatic

EM plane wave type solutions of the general form:

















, , , , , , , , , , i t o i t o E x y z t E x y z e B x y z t B x y z e           _ _

Maxwell’s Equations (inside the rectangular resonant cavity – away from the walls):

(1) Gauss’ Law: (2) No Monopoles:

 E0   E_o 0  B0   B_o 0

(3) Faraday’s Law: (4) Ampere’s Law:

E B t          Eo i B o  _ _ B 1₂ E c t         Bo i ₂ Eo c      

Take the curl of (3): = 0 {Gauss’ Law}

 

 

 

2 2 o o o o o E i B E E i B E c                 _{  }     _  _   _ _  _ _

 {using (4) Ampere’s Law}

 2 2 2 2 2 2 ox ox oy oy oz oz E E c E E c E E c       _{  }      _{  }      _{  }    _  _  

















, , , , . . each is a fcn , , , , ox ox oy oy oz oz E E x y z E E x y z i e x y z E E x y z       _                

Subject to the boundary conditions

|| 0

E  andB_ 0 at all inner surfaces.

For each component



x y z, ,



of E_o



x y z, ,



we try product solutions and then use the separation of variables technique:



, ,



     

i o i i i E x y z X x Y y Z z for





2 2 _{, ,} i i o o E x y z E c     _{  }     _{where subscript ix y z, , .}

   

2 2

 

   

2 2

 

   

2 2

 

2

     

i i i i i i i i i i i i X x Y y Z z c Y y Z z X x Z z X x Y y X x Y y Z z x y z c     _{ }   _{  }     

Divide both sides by Xi

     

x Y y Z zi i : The wave equation becomes:

 

 

 

 

 

 

 

 

  2 2 2 2 2 2 2

only only only

1 i 1 i 1 i i i i fcn x fcn y fcn z X x Y y Z z X x x Y y y Z z z c        _{ }   _{  }        

This equation must hold/be true for arbitrary (x, y, z) pts. interior to resonant cavity 0 0 0 x a y b z d      _{ }     _{ }   

This can only be true if:

 

 

2 2 2 1 constant i x i X x k X x x      

 

_{ }

2 2 2 0 i x i X x k X x x    

 

 

2 2 2 1 constant i y i Y y k Y y y      

 

_{ }

2 2 2 0 i y i Y y k Y y y    

 

 

2 2 2 1 constant i z i Z z k Z z z      

 

_{ }

2 2 2 0 i z i Z z k Z z z     with: 2 2 2 2 2 x y z k k k k c       _{  }    characteristic equation

General solution(s) are of the form:



ix y z, ,



:



, ,



cos

 

sin

 

cos

 

sin

 

cos

 

sin

 

i

o i x i x i y i y i z i z

E x y z _A k x B k x __C k y D k y __E k z F k z _ n.b. In general, , and k k_{x y} k_z should each have subscript ix y z, , , but we will shortly find out that

i

x

k  same for all , , ,

i

y

ix y z k  same for all ix y z, , , and

i

z

k  same for all ix y z, , . n.b. We seek

oscillatory

(not damped)

Boundary Conditions: E_||0 @ boundaries and B_ 0 @ boundaries: 0, 0 at 0, ox y y b E z z d      _{ }     coefficients 0 0 x x E C         and , 1, 2,3,.... , 1, 2,3,.... y z k n b n k d               0, 0 at 0, oy x x a E z z d      _{ }     coefficients 0 0 y y A E         and , 1, 2,3,.... , 1, 2,3,.... x z k m b m k d        _{ }      0, 0 at 0, oz x x a E y y b      _{ }     coefficients 0 0 z z A C         and , 1, 2,3,.... , 1, 2,3,.... x y k m a m k n d n          _{ }     

n.b. m = 0, and/or n = 0 and/or 0are not allowed, otherwise



, ,



0

i

o

E x y z  (trivial solution).

Thus we have (absorbing constants/coefficients, & dropping x,y,z subscripts on coefficients):





 

 

 

 





 

 

 

 





 

 

 

 

, , cos sin sin sin

, , sin cos sin sin

, , sin sin cos sin

ox x x y z oy x y y z ox x y z z E x y z A k x B k x k y k z E x y z k x C k y D k y k z E x y z k x k y E k z F k z   _  _    _  _    _  _         

But (1) Gauss’ Law:  E0  Eox Eoy Eoz 0

x y z   _{ } _       Thus:

 

 

 

 

 

 

 

 

 

 

 

 

sin cos sin sin

sin sin cos sin

sin sin sin cos 0

x x x y z y x y y z z x y z z k A k x B k x k y k z k k x C k y D k y k z k k x k y E k z F k z         _  _    _  _      

This equation must be satisfied for any/all points inside rectangular cavity resonator.

In particular, it has to be satisfied at



x y z, ,







0,0,0



.

We see that for the locus of points associated with (x = 0,y,z) and (x,y = 0,z) and (x,y,z = 0), we must have BD  F 0 in the above equation.

Note also that for the locus of points associated with



xm 2 , ,k_x y z



and



x y, n 2 ,k_y z



and



x y z, ,  2kz



where , ,m n odd integers (1, 3, 5, 7, etc. …) we must have:

0

x y z

Ak Ck Ek  .

Thus:





 

 

 





 

 

 





 

 

 

1, 2,3, 4,

, , cos sin sin

, , sin cos sin 1, 2,3, 4,

, , sin sin cos

1, 2,3, 4, x ox x y z oy x y z y oz x y z z m k m a E x y z A k x k y k z n E x y z C k x k y k z k n b E x y z E k x k y k z k d     _  _    _{ }   _      _ _{ }      _{  }    _{ }         _     

With: E_o



x y z, ,



E x_oxˆE y_oyˆE z_ozˆ n.b.: m  n  0 simultaneously is not allowed!

Now use Faraday’s Law to determine B:





i B E       

 

 

 

 

 

 

sin cos cos sin cos cos

oy oz ox y x y z z x y z E E i i B Ek k x k y k z Ck k x k y k z y z      _{ }   _  _  _  _         

 

 

 

 

 

 

cos sin cos cos sin cos

ox oz oy z x y z x x y z E E i i B Ak k x k y k z Ek k x k y k z z x      _{ }   _  _  _  _         

 

 

 

 

 

 

oy ox cos cos sin cos cos sin

oz x x y z y x y z E E i i B Ck k x k y k z Ak k x k y k z x y      _{ }   _  _   _  _          or:









 

 

 







 

 

 





 

 

 



ˆ ˆ ˆ , , ˆ

sin cos cos

ˆ

cos sin cos

ˆ

cos cos sin

o ox oy oz y z x y z z x x y z x y x y z B x y z B x B y B z i Ek Ck k x k y k z x Ak Ek k x k y k z y Ck Ak k x k y k z y             _{  }      

This expression for B_o



x y z, ,



(already) automatically satisfies boundary condition (2) B_ 0: 0 ox B  at 0,x xa Boy 0at 0,y yb Boz 0at 0,z zd with k_x m a         with y n k b         with kz d          m0,1, 2, n0,1, 2, 0,1, 2,

Does  B_o



x y z, ,



0 ???











 

 

 

, ,

cos cos cos

oy ox oz o x y z x y z x y B B B B x y z x y z i i k Ek Ck k x k y k z k k E                      _  



 x z k k C  





 

 

 

k_y Ak _z Ek _x cos k x_x cos k y_y cos k z_z  k k A_{y z}   k k E_{x y}





 

 

 



k Ck_z  _xAk _y cos k x_x cos k y_y cos k z_z k k C_{x z}   k k A_{y z} 



 

 

 

cos  k x_x cos k y_y cos k z_z _





B x y z

o



, ,





0

 

_

_YES!!!

For TE Modes: 0

z

E   coefficient E 0. Then Ak _xCk _yEk _z 0 tells us that: Ak _xCk _y 0 or: x y k C A k      _{ }     Thus:



, , ,



cos

 

sin

 

sin

 

i t

ox x y z E x y z t  A k x k y k z e k_x m ,m 1, 2,3, a    _{ }    



, , ,



x sin

 

cos

 

sin

 

i t

oy x y z y k E x y z t A k x k y k z e k        _{ }     _{, 1, 2,3,} y n k n b    _{ }     E_oz



x y z t, , ,



0 k_z , 1, 2,3, d    _{ }      

(n = 0 is NOT allowed for TE modes!!!)



, , ,



x sin

 

cos

 

cos

 

i t

ox z x y z y k i B x y z t A k k x k y k z e k        _ _    



, , ,



cos

 

sin

 

cos

 

i t

oy z x y z i B x y z t k A k x k y k z e       



, , ,



x cos

 

cos

 

sin

 

i t

oz x y x y z y k i B x y z t A k k k x k y k z e k         _ _   _{ }     

The angular cutoff frequency for _{, ,} th

m n  mode for TE modes in a rectangular cavity is:

2 2 2 mn m n c a b d      _ _ _ _ _ _         and: vg c v k      no dispersion. The lowest , , m n TE _ mode



a b d



is: TE₁₁₁

For TM Modes: 0 z B  



Ck _xAk _y



0 or: y x k C A k           Thus:



, , ,



cos

 

sin

 

sin

 

i t

ox x y z E x y z t  A k x k y k z e k_x m ,m 1, 2,3, a    _{ }    



, , ,



y sin

 

cos

 

cos

 

i t

oy x y z x k E x y z t A k x k y k z e k      _{ }     _{, 1, 2,3,} y n k n b    _{ }    

(m = 0 is NOT allowed for TM modes!!!) kz , 1, 2,3, d    _{ }      



, , ,



x y y sin

 

sin

 

cos

 

i t

oz x y z z x z k k k E x y z t A k x k y k z e k k k          _{ }_{  }         



, , ,



y y y sin

 

cos

 

cos

 

i t

ox x y z x y z x z x k k k i B x y z t A k k A k k x k y k z e k k k                 _{ } _{   } _{  }               



, , ,



y x cos

 

sin

 

cos

 

i t

oy z x y x y z x x k _k i B x y z t Ak A k k k x k y k z e k k              _  _ _{   }              B_oz



x y z t, , ,



0

For either TE or TM modes:

2 2 2 2 2 x y z k k k k c       _{  }   with: , 1, 2, x m k m a    _{ }     y , 1, 2, n k n b    _{ }     kz , 1, 2, d    _{ }      

The angular cutoff frequency for _{, ,} th

m n  mode is the same for TE/TM modes in a rectangular cavity:

2 2 2 mn m n c a b d      _ _ _ _ _ _         and: vg c v k      no dispersion. The lowest , , m n TM _ mode



a b d



is: TM₁₁₁

B.) The Spherical Resonant Cavity:

The general problem of EM modes in a spherical cavity is mathematically considerably more involved (e.g. than for the rectangular cavity) due to the vectorial nature of the E and B-fields.

For simplicity’s sake, it is conceptually easier to consider the scalar wave equation, with a

scalar field 

 

r t, satisfying the free-source wave equation

 

 

2 2 2 , 1 , r t 0 r t c t         

which can be Fourier-analyzed in the complex time-domain 

 

r t, 



r,



ei td





_

 

with each Fourier component 



r,



satisfying the Helmholtz wave equation:



2 2





_,



₀ k  r 

    with: 2





2

k   c i.e. no dispersion.

In spherical coordinates, the Laplacian operator is:





2













2





2 2 2 2 2 2 , , 1 1 1 , , sin sin sin r r r r r r r r r                        _{ }   _  _     

To solve this scalar wave equation – we again try a product solution of the form:



,



R r

     

i t r P Q e r     _    _

  



, spherical harmonics , m m m f r Y  



   

Plug this 



r,



into the above scalar wave equation, use the separation of variables technique:

Get radial equation:





 

2 2 2 2 1 2 0 d d k f r dr r dr r               where: = 0, 1, 2, 3, . . . Let: f

 

r 1 u r

 

r 

  . Then we obtain Bessel’s equation with index v  12 :





2

_{ }

2 1 2 2 2 2 1 0 d d k u r dr r dr r  _             

Solutions of the (radial) Bessel’s equation are of the form:

 

1

 

1

 

2 2

1 1

2 2

Bessel fcn of 1st Bessel fcn of 2nd kind of order kind of order

m m m A B f r J kr N kr r  r              

It is customary to define so-called spherical Bessel functions and spherical Hankel functions:

 

 

1 2 1 2 2 j x J x x       _{ }   where: xkr

 

 

1 2 1 2 2 n x N x x       _{ }   The Ym



 ,



satisfy the angular portion of scalar wave equation…

and:  

 

 

 

 

 

1 2 1 1 2 2 1,2 2 h x J x iN x j x i n x x     _{  } _{  }  _        

 

 

1 d sin

 

x j x x x dx x       _{   }  _{  }   

 

 

1 d cos

 

x n x x x dx x        _{   }  _{  }   

 

 

0 sin x j x x   1

 

0 ix e h x ix 

 

 

0 cos x n x x    2

 

0 ix e h x ix   

 

 

 

1 2 sin cos x x j x x x   1 1

 

1 ix e i h x x x     _  _  

 

 

 

1 2 cos x sin x n x x x    1 2

 

1 ix e i h x x x  _{ }   _  _   For x1, :

 

_

_

1

_

2

_

... 2 1 !! 2 2 3 x x j x  _   _  _  _     where:



21 !!

 

 21 2



1 2



3 ... 5 3 1



  

 



2 ₁1 !!



1

_

2

_

... 2 1 2 x n x x       _   _        For x1, :

 

sin 1 2 j x x x     _  _    

 

1cos 2 n x x x      _  _    

The general solution to Helmholtz’s equation in spherical coordinates can be written as:

 

  1  1

 

 2  2

 





, , m m m , m r t A h kr A h kr Y           



      

Coefficients are determined by boundary conditions. For the case of EM waves in a spherical resonant cavity we will (here) only consider TM modes, which for spherical geometry means that the radial component of B B, 0_r  . We further assume

(for simplicity’s sake) that the E and B-fields do not have any explicit -dependence.

n.b. If x = kr is real, then

 2

_{ }

* 1 

_{ }

Hence:

















2 1 ! , cos 4 ! m im m m Y P e m               

Will have some restrictions imposed on it Associated Legendré Polynomial If Br 0 and B



explicit function of , then:  B0  B_ 0 {necessarily}

But: E B t         requires: E_ 0

 TM modes with no explicit -dependence involve onlyE_r, E_ and B_

Combining E B t         and B 1₂ E c t       

with harmonic time dependence i t

e of solutions, We obtain: 2 0 B B c    _{  }        

The -component of this equation is:

 

 





2 2 2 2 1 1 sin 0 sin rB rB rB c  r  r   _         _{ }  _   _{ }  _      But:





 

₂

~Legendré equation with 1

1 1

sin sin

sin sin sin

m rB rB rB_                   _  _{ }   _{ }      _  _ 

 Try product solutions of the form: B

 

r, u r

   

P1 cos r

     

Substituting this into the above equation gives a differential equation for u r_

 

of the form of:

Bessel’s equation:

 



  

2 2 2 2 1 0 d u r u r dr c r  _{ }   _{ }  _           

with = 0, 1, 2, 3, . . . defining the

angular dependence of the TM modes.

Let us consider a resonant spherical cavity as two concentric, perfectlyconducting spheres of

inner radius a and outer radius b. If

 

_, u r

   

1 _cos

B r P

r

      , the radial and tangential electric fields (using Ampere’s Law) are:

 

, 2



sin



2



1

   

cos



sin r u r ic ic E r B P r  r r                 

 

_, ic2

 

ic2 u r

   

1 _cos E r rB P r r r r   _  _           

But E_ E_ which must vanish at r = a and r = b 

 

r a

 

r b 0 u r u r r  r         

The solutions of the radial Bessel equation are spherical Bessel functions (or spherical Hankel functions).

The above radial boundary conditions on

 

_{r a} 0 r b u r r      _{lead to}

transcendental equations for

the characteristic frequencies,  {eeeEEK}!!!

However {don’t panic!}, if: (b – a) = h is such that ha then:



₂ 1





₂ 1



constant!!!

r a

 

 

   

And thus in this situation, the solutions of Bessel’s equation:

 

2





_{ }

2 2 2 1 0 d u r u r dr c a  _{ }   _{ }              

 

 

2 2 2 0 d u r k u r dr     where:





2 2 2 1 k c a     _{ }     

are simply sin (kr) and cos (kr) !!! i.e. u r_

 

 Acos

 

kr Bsin

 

kr

Then: sin

 

cos

 

0

r a u kA ka kB ka r       

 _{and sin}

 

_cos

 

₀

r b u kA kb kB kb r        

For



b a



ha an approximate solution is: u r_

 

 Acos



krka



with: khk b a







n , n = 0, 1, 2, . . . Thus:





2 2 2 2 1 n n k c a h        _{ }  _{ }        , n = 0, 1, 2, 3, . . . and = 0, 1, 2, 3, . . . The corresponding angular cutoff frequency is:





2





2 2 2 1 1 n n n c k c a h a     _ _            for ha, n = 0, 1, 2, 3, . . . and = 0, 1, 2, 3, . . .

Because ha, we see that the modes with n = 1, 2, 3, . . . turn out to have relatively high frequencies n n c h   _{ } _  

 for n1. However, the n = 0 modes have relatively low frequencies:





_

_

0 2 1 1 c c a a           for ha.

An exact solution (correct to first order in (h/a) expansion) for n = 0 is:



 



0 ₁ 2 1 c a h       

These eigen-mode frequencies are known as Schumann resonance frequencies. = 1, 2, 3, . . .

For n = 0, the EM fields are: E_0 0, E_r0 1₂ P



cos



r     and 0 1 1



_cos



B P r    

Very Useful Table:

ˆ _ˆ ˆ r   ˆ  rˆ ˆ ˆ ˆ ˆr     ˆ  ˆ rˆ ˆ ˆ ˆr   rˆ  ˆ ˆ Poynting’s vector:











 

1





 

1



 

0 0 0 3 3 0 1 _{ˆ ˆ} 1 1 _ˆ

cos cos cos cos

S E B r P P P P r r                        

The Earth’s surface and the Earth’s ionosphere behave as a spherical resonant cavity (!!!) with the Earth’s surface {approximately} as the inner spherical surface: ar_ r_ 6378 km

6 6.378 10 m

  (= Earth’s mean equatorial radius), the height h (above the surface of the Earth) of the ionosphere is: _{100 10}5

h km m(a) → b = a + h 6.478 x 106m. For the n = 0 Schumann resonances:



 



0 ₁ 2 1 c a h        for ha. 1:  





01 ₁ 2 2 c a h     01 01 10.5 2 f  Hz    2 :  





02 ₁ 2 6 c a h     02 02 18.3 2 f  Hz    3:  





03 ₁ 2 12 c a h     03 03 25.7 2 f  Hz    4 :  





04 ₁ 2 20 c a h     04 04 33.2 2 f  Hz    5 :  





05 ₁ 2 30 c a h     05 05 46.7 2 f  Hz    (. . . etc.)

The n = 0 Schumann resonances in the Earth-ionosphere cavity manifest themselves as peaks in the noise power spectrum in the VLF (Very Low Frequency) portion of the EM spectrum → VLF EM standing waves in the spherical cavity of the Earth-ionosphere system!!!

ˆ y ˆ z ˆ x  ˆ  0 ˆ,B_ ˆ   0 ˆ, r ˆ r E r  0 S  a b Circumpolar N-S waves! n.b. For the n = 1 Schumann resonances: 1 1.5 f  KHz

Schumann resonances in the Earth-ionosphere cavity are excited by the radial E-field component of lightning discharges (the frequency component of EM waves produced by lightning at these Schumann resonance frequencies).

Lightning discharges (anywhere on Earth) contain a wide spectrum of frequencies of EM radiation – the frequency components f01, f02, f03, f04, . . excite these resonant modes – the Earth literally “rings like a bell” at these frequencies!!! The n = 0 Schumann resonances are the lowest-lying normal modes of the Earth-ionosphere cavity.

Schumann resonances were first definitively observed in 1960. (M. Balser and C.A. Wagner, Nature 188, 638 (1960)).

 Nikola Tesla may have observed them before 1900!!! (Before the ionosphere was known to

even exist!!!) He also estimated the lowest modal frequency to be f01 ~ 6 Hz!!!

The observed Schumann resonance frequencies are systematically lower than predicted, (primarily) due to damping effects: 2 ₀2 1 1 i

Q

  _   _

  where Q = Quality factor

0





 = “Q”

of resonance, and  = width at half maximum of power spectrum: The Earth’s surface is also not perfectly conducting.

Seawater conductivity C 0.1 Siemens!!

Neither is the ionosphere! → Ionosphere’s conductivity ₁₀ 4 ₁₀ 7

C

 _  _  _Siemens

 On July 9, 1962, a nuclear explosion (EMP) detonated at high altitude (400 km) over

Johnson Island in the Pacific {Test Shot: Starfish Prime, Operation Dominic I}.

- Measurably affected the Earth’s ionosphere and radiation belts on a world-wide scale!

- Sudden decrease of ~ 3 – 5% in Schumann frequencies – increase in height of ionosphere!

- Change in height of ionosphere:   h h h2 0.03 0.05

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R_ 400 600  km !!!

- Height changes decayed away after ~ several hours. - Artificial radiation belts lasted several years!

 Note that # of lightning strikes, (e.g. in tropics) is strongly correlated to average temperature.

Scientists have used Schumann resonances & monthly mean magnetic field strengths to monitor lightning rates and thus monitor monthly temperatures – they all correlate very well!!!

 Monitoring Schumann Resonances → Global Thermometer → useful for Global Warming studies!!

Earth Coordinate System:

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0 ₁ 2 1 2 c f a h        0 ₀ E_  (north – south)

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0 2 1 cos r E P r     (up – down)

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0 1 1 _cos B P r    (east – west)

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1

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0 3 1 _ˆ cos cos S P P r         (north – south)

For the n = 0 modes of Schumann Resonances:

ˆ

We can observe Schumann resonances right here in town / @ UIUC!! Use e.g. Gibson P-90 single-coil electric guitar pickup

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LP₉₀ 10 Henrys, ~10K turns #42AWG copper wire

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for

detector of Schumann waves and a spectrum analyzer (e.g. HP 3562A Dynamic Signal Analyzer) – read out the HP 3562A into PC via GPIB.

 Orientation/alignment of Gibson P-90 electric guitar pickup is important – want axis of

pickup aligned  B ˆ (i.e. oriented east – west) as shown in figure below. n.b. only this

orientation yielded Schumann-type resonance signals {also tried 2 other 90o _orientations {up-down} and {north-south} but observed no signal(s) for Schumann resonances for these.}

Electric guitar PU’s are very sensitive – e.g. they can easily detect car / bus traffic on street below from 6105 ESB (6th _{Floor Lab) – can easily see car/bus signal from PU on a ‘scope!!!}