J. Modern Physics, 2010, 1, 100-107
doi:10.4236/jmp.2010.12015 Published Online June 2010 (http://www.SciRP.org/journal/jmp)
A Second-Order Eigen Theory for Static
Electromagnetic Fields
Shaohua Guo
School of Civil Engineering and Architecture, Zhejiang University of Science and Technology, Hangzhou, China
Received March 28th, 2010; revised May 1st, 2010; accepted May 20th, 2010
Abstract
The static electromagnetic fields are studied here based on the standard spaces of the physical presentation, and the
modal equations of static electromagnetic fields for anisotropic media are deduced. By introducing a set of new poten-
tial functions of order 2, several novel theoretical results were obtained: The classical potential functions of order 1
can be expressed by the new potential functions of order 2, the electric or magnetic potentials are scalar for isotropic
media, and vector for anisotropic media. The amplitude and direction of the vector potentials are related to the anisot-
ropic subspaces. Based on these results, we discuss the laws of static electromagnetic fields for anisotropic media.
Keywords: Anisotropic Media, Static Electromagnetic Field, Standard Spaces, Modal Equation Formatting
1. Introduction
By the Maxwell’s electromagnetic field equations, we
know that the electric and magnetic field are independent
each other under the condition of static fields. The clas-
sical electromagnetic field theory also believes that the
static electric field can be described by a scalar potential
function, and the magnetic field by a vector one. Fur-
thermore, for the passive region, the magnetic field can
also be described by a scalar potential function [1,2].
But it should be pointed out that these results can only be
obtained in the condition of isotropy, and are also only
suit for the isotropic media. However, with the develop-
ment of material science, more and more anisotropic
dielectric or magnetic materials are applied to various
fields, such as electron devices, communications and
sensors, even for the traditional geological structure, we
also can see the electrically anisotropic media or mag-
netically anisotropic media. It is found by recent re-
search works that the limitations of classical static elec-
tromagnetic field theory have become obvious for these
anisotropic media. For example, the above results for
isotropic media don’t exist for anisotropic media, even
we don’t know the definite form of the electric field po-
tential function or magnetic field potential function,
which make a great difficulty in solving the problem of
anisotropic static electric or magnetic fields [3-5]. Unlike
the classical static electromagnetic field theory, which
studies the Maxwell’s equations under the geometric
representation, in this paper, the Maxwell’s equations are
restudied under the physical representation. As the result
of this, the modal equations of static electric or magnetic
fields are deduced, which give the novel expressions for
the potential functions of static electric or magnetic
fields for anisotropic media, and bring to light the intrin-
sic laws of static electromagnetic field.
2. Standard Spaces of Electromagnetic
Media
In anisotropic electromagnetic media, the dielectric per-
mittivity and magnetic permeability are tensors instead
of scalars. The constitutive relations are expressed as
follows
D
E


 (1)
B
H


 (2)
where the dielectric permittivity matrix

and the
magnetic permeability matrix

are usually symmetric
ones, and the elements of the matrixes have a close rela-
tionship with the selection of reference coordinate. Sup-
pose that if the reference coordinates is selected along
principal axis of electrically or magnetically anisotropic
media, the elements at non-diagonal of these matrixes
turn to be zero. Therefore, Equations (1) and (2) are called
the constitutive equations of electromagnetic media un-
der the geometric presentation. Now we intend to get rid
of effects of geometric coordinate on the constitutive

A Second-Order Eigen Theory for Static Electromagnetic Fields 101
equations, and establish a set of coordinate-independent
constitutive equations of electromagnetic media under
physical presentation. For this purpose, we solve the fol-
lowing problems of eigen-value of matrixes

-0Ij 

(3)

-0I  

(4)
where

1, 2, 3
ii
  and


1, 2, 3
i
gi are respectively
eigen dielectric permittivity and eigen magnetic perme-
ability, which are constants of coordinate-independent.

1, 2, 3
i
ji and


1, 2, 3
ii
  are respectively eigen
electric vector and eigen magnetic vector, which show
the electrically principal direction and magnetically prin-
cipal direction of anisotropic media, and are all coordi-
nate-dependent. We call these vectors as standard spaces.
Thus, the matrix of dielectric permittivity and magnetic
permeability can be spectrally decomposed as follows


 (5)


 (6)
where


123
,,diag

  and


123
,,diag

 
are the matrix of eigen dielectric permittivity and eigen
magnetic permeability, respectively.

123
,,jjj and

123
,,

  are respectively the modal matrix of
electric media and magnetic media, which are both or-
thogonal and positive definite ones, and satisfy
T
I
 ，T
I
 .
Projecting the electromagnetic physical qualities of the
geometric presentation, such as the electric field intensity
vector E, magnetic field intensity vector
H
, magnetic
flux density vectorB and electric displacement vec-
tor D, into the standard spaces of the physical presenta-
tion, we get
*1, 2, 3
ii
DjDi

 (7)
*1, 2, 3
ii
EjEi

 (8)
*1, 2, 3
ii
BBi 

 (9)
*1, 2,3
ii
HHi 

 (10)
These are the electromagnetic physical qualities under
the physical presentation.
Substituting Equations (7)-(10) into Equations (1) and
(2) respectively, and using Equations (5) and (6) yield,
we have




**
DE (11)




**
BH (12)
or
** 1, 2, 3
iii
DEi  
(13)
** 1, 2, 3
iii
BHi  
(14)
The above equations are just the modal constitutive
equations in the form of scalar.
3. Matrix form of Static Electromagnetic
Field Equation
The classical static Maxwell’s equations in passive re-
gion can be written as
=0, =0ED


(15)
=0, =0
H
B

 (16)
where

is a Hamilton operator. It is seen from the
above equations that the electric field and magnetic field
are not only independent, but also the same in the form
of equation. So, it is undistinguishable to study the prob-
lems of electric field or magnetic field under the static
condition. For this purpose, we consider here only the
problem of electric field.
From Equation (15), we can see that one is a vector
equation, another is scalar one. It is well known that the
vector equation can be written as the matrix one, but the
scalar equation can not. By the first one of Equation (15),
we have




0E

 (17)
where

0
0
0
z
y
z
x
yx







 




 


(18)
It is an operator matrix of order 1.
In order get the matrix expression of static electromag-
netic equations, both of Equation (15) should be re-
formed in a suitable form.
For dynamic electromagnetic fields, a matrix equation
of electromagnetic waves be dedued by author [6]



 




2
t
EE   (19)
where
 




z
zyy xyxz
yxxx zzyz
zx zyxx yy









 







(20)
It is an operator matrix of order 2. For static electro-
mag-netic fields, we have




0E

 (21)
Now, rewriting the second one of Equation (15) in the

A Second-Order Eigen Theory for Static Electromagnetic Fields 102
index form of tensor
'0 ii
D (22)
Differentiating the above equation with index
j
, it
become a vectorial one
'0 iij
D (23)
Rewritting it in the matrix form, we get



0D
(24)
where

11 21 31
12 22 32
13 2333



 




(25)
In this paper,


 and


 is defined as the matrix
of electic intensity and electic displacement operators
respectively
4. Eigen Equations of Static Electric Fields
Now, we transform the matrix equations of static electric
field into modal ones.
Substituting Equation (7) into Equation (21), and mul-
tiplying it with the transpose of modal matrix in left, we
have



*0
TE (26)
It be proved [6] that the matrix of electric intensity
operator can also be spectrally decomposed, that is

*
T


(27)
Thus, Equation (26) can be uncoulped and become

**
0E


 (28)
or
** 01,2,3
ii
Ei (29)
in which

** *1, 2, 3
T
ii i
i  (30)
where

  
*T
ii i

  (31)
In same way, substituting Equation (7) into Equation
(24), and multiplying it with the transpose of modal ma-
trix in left, we have



*0
TD (32)
let

*
T


(33)
and substituting Equation (11) into Equation (32),we
have




**
0E



 (34)
Comparing Equation.(34) with Eq.(28) ,we get


**






(35)
It is seen that *




 is also a diagonal matrix. We call
it as eigen matrix of electric displacement operator. Thus,
we have
** 01,2,3
ii
Di (36)
So, Equations (29) and (36) constitute of the eigen
equations of static electric field. Different from the clas-
sical ones, they show the simplicity and symmetry of
static electromagnetic law.
5. General Solution of Eigen Equations of
Static Electric Fields
Let
** 1,2, 3
iii
Ei   (37)
** 1, 2, 3
iii
Di   (38)
where



is an unknown row vector, which is new
electric potential function of order 2.
Substituting Equations (37) and (38) into Equations
(29) and (36) respectively, a unified equation are ob-
tained as follows
** 01,2,3
ii i
  (39)
where,


*** 1,2, 3
iii
i  is ith modal operator of
electric field, and a differential operator of order 4. Def-
erent from the Laplce’s equation for the classical electric
potential function of order 1, the new electric potential
function of order 2 can be solved by the modal differen-
tial equation of higher order, and the classical electric
potential function of order 1 can be expressed by the new
electric potential function of order 2. Once the modal
potential functions are solved from Equation (41), the
electric intensity and electric displacement can be ob-
tained by the following conversion





 
** *
1112223 33

 E (40)







** *
11112 2223 333

   D (41)
In order to get the classical electric potential function
of order 1, we rewritting Equation (40) by using Equa-
tions (30) and (35)













*** *
TT
iiiii ii i
ii

 

E (42)

A Second-Order Eigen Theory for Static Electromagnetic Fields 103
let


*T
iiii

  (43)
It is just the electric potential function of order 1 for
anisotropic media. Thus the electric intensity and electric
displacement can be expressed by the electric potential
function of order 1 as follows

 
** *
11 22 33

 E (44)

 
** *
111 222333





  D (45)
6. The Modal Boundary Condition of Static
Electric Field
It is seen from above that in order to get the solutions of
the electric intensity and electric displacement, we can
turn to solving the modal potential functions. So, the
modal Equation (39) should have the corresponding mo-
dal boundary condition.
An effective boundary case is: Electric displacement
functions of two side of interface should be equal






12
DD (46)
or
 
12 1, 2, 3
ii
DD i (47)
Rewriting Equation (46) in the modal form, we have
 

 

1* 1*2*2*

 

 (48)
or
 
1* 1*2*2*1, 2, 3
iiiii
  (49)
and
 
11*1*22*2*1, 2, 3
ii ii iii
    (50)
7. Application
In this section, we discuss the laws of static electric field
only in anisotropic dielectrics.
7.1. Isotropic Crystal
The matrix of dielectric permittivity of isotropic dielec-
trics is following
00
00
00













(51)
The eigen-values and eigen-vectors are respectively
shown as below


11 11 11
,,diag

  
(52)
100
010
001












(53)
We can see from the above equations that there is only
one eigen-space in isotropic crystal, which is a triple-
degenerate one, and the space structure is following


3
1123
,,W

W (54)
The basic vector of one dimension in a triple-degenerate
subspace is

*
1
31,1,1
3
T
  (55)
The eigen electric displacement operator of isotropic
crystal are

* 222
1123
1
3

   (56)


*
1123
3,,
3

 (57)
Therefore, the static electric field equation in isotropic
crystal can be written as below

2
222
10 xyz


  (58)
Thus, the electric strength and electric displacement of
isotropic crystal become
 

*222
1111231
1
1
1
E




 

 (59)
 

* 222
11111112 31
1
1
1








D
(60)
The classical electric potential function of order 1 is


1123i

    (61)
So, the electric intensity and electric displacement of
isotropic crystal can also be expressed by the classical
electric potential function of order 1 as follows

1
x
y
z
E









(62)
or
1




E (63)

A Second-Order Eigen Theory for Static Electromagnetic Fields 104

11 1
x
y
z
D







 






(64)
or
11 1

 

D (65)
It is seen that Equations (62)-(65) are the same as the
classical results, in which the electric potential is a scalar.
But from the following analysis, we will see that only for
isotropy we have same results as classical theory.
7.2. Uniaxial Crystal
The matrix of dielectric permittivity of uniaxial dielec-
trics is following
11
11
33
00
00
00









(66)
The eigen-values and eigen-vectors are respectively
shown as below


11 1133
,,diag

  
(67)
100
010
001






(68)
We can see from the above equations that there are
two eigen-spaces in uniaxial crystal, one of which is a
double-degenerate space, and the space structure is fol-
lowing

 
21
112 23
,WW

 W (69)
The basic vectors in two subspaces are following

*
1
21, 1,0
2
T
  (70)

*
20, 0,1T
  (71)
The eigen electric strength qualities of uniaxial crystal
are respectively shown as below
*T
22 3
EE 
E
= (72)
T* T*
11 22
EE  E (73)
Multiplying Equation (57) with 2

, using T
21
0

 
and

T11,2
ii i 

, we get

T
*T*T*22
1222212
E
EEEEEE 
(74)
The eigen electric displacement operators of uniaxial
crystal are respectively shown as below

*
11122
1
2

 (75)
*
233

 (76)


*
112
2,,0
2
 (77)




*
23
0, 0,

 (78)
Therefore, the static electric field equation in uniaxial
crystal can be written as below

2
22 *
10 xy


 (79)
4*
20 z


 (80)
It is seen from Equations (79) and (80) that there are
two static electric fields in uniaxial crystal. Thus, the
electric intensity and electric displacement become









**
1112 22
1122133 2
*
1122 1
*
1122 1
*
33 2
10
10
01
E  






 
 

 
 
 










(81)






**
11112222
*
11 11221
*
11 11221
*
33 332
D

 



 










(82)
The classical electric potential function of order 1 is

1121
1
2

  
(83)
132



 (84)
So, the electric intensity and electric displacement of
uniaxial crystal can also be expressed by the classical
electric potential function of order 1 as follows

11
21
32
E








 






(85)

11 1
11 1
33 2
x
y
z
D







 


 






(86)
It is seen that the electric intensity and electric dis-
placement of uniaxial crystal are quite different from

A Second-Order Eigen Theory for Static Electromagnetic Fields 105
those in isotropic crystal, and there exist two kinds of
modal electric potential functions in uniaxial crystal, so
they become vectorial ones, this is also different from the
classical results of static electric field.
7.3. Biaxial Crystal
The matrix of dielectric permittivity of biaxial dielectrics
is following
11
22
33
00
00
00









(87)
The eigen-values and eigen-vectors are respectively
shown as below


11 2233
,,diag

  
(88)
100
010
001






(89)
We can see from the above equations that there are
three eigen-spaces in biaxial crystal, and the space struc-
ture is following






11 1
1122 33
WWW

W (90)
The eigen-qualities and eigen electric displacement
operators of biaxial crystal are respectively shown as
below
*T
11 1
EE 
E
= (91)
*T
22 2
EE 
E
= (92)
*T
333
EE 
E
= (93)



**
111 1 1
,0,0T  (94)



**
222 22
0,, 0T  (95)



**
333 33
0, 0,T  (96)
Therefore, the static electric field equation in biaxial
crystal can be written as below
4*
10 x
  (97)
4*
20 y
  (98)
4*
30 z
  (99)
It is seen from Equations (97)-(99) that there are three
static electric fields in biaxial crystal. Thus, the electric
intensity and electric displacement become





 
** *
1112 22333
** *
11 22 33
11 1
22 2
33 3
100
01 0
001
E

 





 
 

 
 
 










(100)

11 111
22 222
33 333z
D




















(101)
The classical electric potential function of order 1 is
111



 (102)
222



 (103)
332



 (104)
So, the electric intensity and electric displacement of
uniaxial crystal can also be expressed by the classical
electric potential function of order 1 as follows

11
22
33
E















(105)

11 1
22 2
33 3
x
y
z
D





 


 






(106)
It is seen that the electric intensity and electric dis-
placement of biaxial crystal are quite different from
those in isotropic crystal, and there exist three kinds of
modal electric potential functions in uniaxial crystal, so
they become vectorial ones, this is also different from the
classical results of static electric field.
7.4. Monoclinic Crystal
The matrix of dielectric permittivity of monoclinic di-
electrics is following
11 12
12 22
33
0
0
00














(107)
The eigen-values and eigen-vectors are respectively
shown as below


1233
,,diag

  
(108)

A Second-Order Eigen Theory for Static Electromagnetic Fields 106



T
121 11
2212
11112
T
122 11
2212
211 12
T
,1,0
1,, 0
0, 0,1





 

















 














(109)
where


2
11 222
1, 2112212
1
22







333

 (110)
We can see from the above equations that there are
also three eigen-spaces in monoclinic crystal, and the
space structure is following






11 1
1122 33
WWW

 W (111)
The eigen-qualities and eigen electric displacement
operators of monoclinic crystal are respectively shown as
below
 
*T
11 1111122
22
11112
1
EEE

 
 


 E=
(112)
 
*T
2212 12112
22
211 12
1
EEE

 
 


 E=
(113)
*T
33 3
EE 
E
= (114)


*2 2
1111122 1112
*
1 11121112
2
,,0
T
ab ab
abab


 


(115)


*22
2211222 2212
*
2 21222122
2
,,0
T
ab ab
abab


 


(116)


**
333 33
0, 0,T  (117)
where

11
12
2212
11 12
1, 2
i
iii
i
babi
 

 



Therefore, the static electric field equation in mono-
clinic crystal can be written as below

2
22 *
1111 2211121
20ab ab    (118)

2
22 *
2112 2222122
20ab ab 

  (119)
4*
30 z


 (120)
It is seen from Equations (118)-(120) that there exist
also three static electric fields in monoclinic crystal,
which is a little different from the results in biaxial crys-
tal because of the distortion of static electric fields. Thus,
the electric intensity and electric displacement become








***
1112223 33
111 2
22
11121 1112211121
2
0
E
ab
ab abab



 
 


  




21 22
22
21222112 2222122
33 3
2
0
0
0
1
ab
ab abab


 


 









(121)









***
11112 2223 333
111 2
22
11 121111 12211121
21 22
22
212222112 2222122
3333
2
0
2
0
0
0
1
D
ab
aba bab
ab
aba bab
 





 
 


  



 


 









(122)
The classical electric potential function of order 1 is

 
111112111121
aa bba b

 



 (123)

 
222122221222
aa bbab

  



 (124)
333


 (125)
So, the electric intensity and electric displacement of
monoclinic crystal can also be expressed by the classical
electric potential function of order 1 as follows

111 22122
111 212 12 223
3
0
0
00
aba b
Eabab


 
 



 
 

 

 

  
(126)

A Second-Order Eigen Theory for Static Electromagnetic Fields 107

11 12
11112 1
21 22
221222333
3
0
0
0
0
ab
Dab
ab
ab


 
 


 











 
(127)
It is seen that the electric intensity and electric dis-
placement of biaxial crystal are aiso quite different from
those in isotropic crystal.
8. Conclusions
In this paper, we construct the standard spaces under the
physical presentation by solving the eigen-value problem
of the matrixes of dielectric permittivity and magnetic
permeability, in which we get the eigen dielectric per-
mittivity and eigen magnetic permeability, and the cor-
responding eigen vectors. The former are coordinate-
independent and the latter are coordinate-dependent. Be-
cause the eigen vectors show the principal directions of
electromagnetic media, they can be used as standard
spaces. Based on the spaces, we get the modal equations
of static electromagnetic fields by converting the classi-
cal Maxwell’s vector equation to the eigen Maxwell’s
scalar equation, each of which shows the existence of an
static electromagnetic field. For example, there is only
one kind of static electromagnetic field in isotropic crys-
tal, which is identical with the classical result; there are
two kinds of static electromagnetic fields in uniaxial
crystal; three kinds of static electromagnetic fields in
biaxial crystal and three kinds of distorted static electro-
magnetic fields in monoclinic crystal. All of these new
theoretical results need to be proved by experiments in
the future.
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[3] M. Zuniga and J. Kong, “Modified Radiative Transfer
Theory for a Twolayer Random Medium,” Journal of
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Strong Anisotropic Inhomogeneities,” Radiophysics and
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[5] K. Chari and P. Silvester, “Finite Elements in Electric
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