Special terminology in this article:
{} indicates superscript (including the power).
[] indicates subscript.
~A means the vector A.
^a means the unit vector a.
<four> means 4.
(A)[av] means the average of A.
In a capital Greek letter the word "cap." is written.
We show integral around a closed space as <circulation>.
We show round d (as used in partial derivation) as <round>.

13          Electromagnetic theory from a new viewpoint
Ed 07.08.01 -------------------------------------------
Abstract
--------
We consider point magnetic charges as the sources of the magnetostatic
fields, like the point electric charges for the electrostatic fields.
Forms of the mutual effects of electric and magnetic charges on themselves
and on each other are presented in the forms of vectorial relations. Using
these relations incorrectness of a usual manner which eventually leads to 
the deviation from the classical physics and to the rejection of the 
Galilean transformations and to the resort to the special relativity is 
proven. Static potential energy of a disribution of electric and magnetic 
charges is presented with a careful view on the actual essence of each
involved term; this itself shows a sample of the usual carelessness
existing in the present current electromagnetic theory even in its static
discussions. Almost all the fundamental relations in the present current 
electromagnetic theory are rewritten in new forms by using the fundamental
vectorial relations presented at the beginning of the article. In a more
detailed argument the proportion of the curl of the dynamic field of one kind
(ie magnetodynamic or electrodynamic) to the time derivative of the static 
field of the other kind (ie electrostatic or magnetostatic) is established;
meanwhile the proportion of the current density of one kind to the 
time derivative of the field of the same kind is also shown. Lenz's law
is obtained in its new form. Static and dynamic inductances are presented.
By presenting an aspect which views the space full of much tiny
electrostatic and magnetostatic dipoles, the possibility of the proportion
of the static fields to the dynamic fields is shown.

The way in which the electromagnetic wave propagates through these dipoles
is easily explained by using the mentioned fundamental relations, and by
obtaining the new form of Maxwell's equations and deducing the wave equations
from them, this simple explanation is endorsed. By deducing 
the dynamic potential energy and explaining its difference with the static
potential energy of a set of charges, the Poynting vector is obtained in 
its new form. It is shown that the fields of an electromagnetic wave are
continuous across the boundary interfaces. Fresnel coefficients are obtained
in their quite new forms, and it is explained that the coefficient appearing
in the fundamental relations showing the relations between two electric
and magnetic charges moving relative to each other, <mu>, must be construed
as a world constant. The reflectance and transmittance are introduced in this 
new approach, and it is shown that the sum of them is identical with one.

I. Introduction
---------------
We know that the present current electromagnetic theory, so far it 
concerns the discussion about electrostatic fields, has often the same
attractiveness as the logical discussions of the mathematics and the
same consistency as the classical mechanics; but from the point where 
the interference of the magnetic and electric fields commences, it becomes
containing not only of some kind of unnatural complexity but also of
sudden unexpected contrariety to the classical mechanics, eg easily and
in simple forms the law of action and reaction is breached by this theory.
It seems that the origin of this problem should be searched in this fact 
that in spite of this fact that the method of simplifying of problems and
in fact the method of subjective modeling of physical problems had been
the usual manner of the great theoretical physicists in discovering the
physical laws, using of this manner for the electromagnetism has been
neglected and instead, much efforts have been made to substantiate some
abstract and subsidiary concepts, like field, and to justify deviation
from the classical mechanics logic mathematically by which unnecessary
complexities of the theory have been increased and consequently the
attractiveness of it has been decreased.

When a high school student reads that the magnetic needle of a compass 
turns beside a wire carrying current and that a force is exerted on a
hanging wire carrying current in a fixed magnetic field, the acceptance
of this matter that this is the same law of action and reaction (and so
there is also some force on the wire due to the compass needle and on the
fixed magnet due to the wire) will be much more logical for him or her 
than ignoring the general validity of this law and attributing the force
exerted on the compass needle directly to some vague field around the wire 
to which he or she should attribute more genuineness than to the agent
causing this field! It is much more logical and desirable for him or her
to visualize two electric and magnetic point charges moving toward 
each other on two different parallel lines and then to deduce the form of 
forces they exert on each other by comparing the situation with the 
mentioned forces exerted on the compass needle and on the hanging wire.
This simple work has been done and in the following section has found its
mathematical and in fact vectorial form. (There you can see the form of the 
force that two moving electric and magnetic charges exert on each other. 
I recommend and emphasize to study the 15th article of this book (after 
study of this (13th) article) for probable physical reason for the existence 
of such form of force.) Just this same simple act has had some interesting 
consequences to which we proceed in this article. What is certain is that 
surely many of the scientists will be glad if some way is found through 
which the electromagnetic theory can be founded totally on the classical 
physics and in the frame of Galilean transformations, since the reason 
presented for the deviation from this physics and the choise of other 
transformations has been the inability to find just such a way.

II. Fundamental relations
-------------------------
We can consider origin in the form of point magnetic chares for the 
magnetostatic fields as we consider origin in the form of point electric
charges for the electrostatic fields. We show the magnetic point charge
by "b" and we assume the signs + (referring to N pole) and - (referring 
to S pole) for the two kinds of magnetic charge. Now with these definitions
if q and q' are two point electric charges and b and b' are two point
magnetic charges and ^r is the outward radial unit vector for each 
charge (see Fig.1), then the following relations will be always true.
In these relations the constant values k, k' and k" are positive.

                  q                  q'
             <----*..................*---->
             ^r[q]                   ^r[q']

    Fig.1. An illustration showing outward radial unit vectors.

Electrostatic force arising from q exerted on q':
           ~F[q'] = kqq'^r[q']/r{2} = q'~E[q']                    (1)

Electrostatic field arising from q in the place of q':
           ~E[q'] = ~F[q']/q' = kq^r[q']/r{2}                     (2)

Magnetostatic force arising from b exerted on b':
           ~F[b'] = k'bb'^r[b']/r{2} = b'~B[b']                   (3)

Magnetostatic field arising from b in the place of b':
           ~B[b'] = ~F[b']/b' = k'b^r[b']/r{2}                    (4)

And when in an inertial reference the point magnetic charge b has a relative
velocity ~v[b] relative to the point electric charge q, then we shall have
the following relations. (It is obvious that ~v[q]=-~v[b]; see Fig.2.)

                  A
                  |      q
                  |      *----->
                  |       ~v[q]'       * b
                  |                    |
INERTIAL REFERENCE|                    | ~v[b]'
                  |                    V
                  .------------------------------>
                 /
               /
             /
           /
        !_

     Fig.2. Particle q has relative velocity ~v[q]=~v[q]'-~v[b]'
            relative to the particle b, and particle b has relative
            velocity ~v[b]=~v[b]'-~v[q]'=-~v[q] relative to the
            particle q.

Electrodynamic force arising from b exerted on q:
~F[q]{*}=k"qb~v[q]<cross>^r[q]/r{2}=(k"/k')q~v[q]<cross>~B[q]     (5)

Electrodynamic field arising from b in the place of q:
~E[q]{*} = ~F[q]{*}/q = k"b~v[q]<cross>^r[q]/r{2} = 
(k"/k')~v[q]<cross>~B[q] = (-k"/k')~v[b]<cross>~B[q]              (6)

Magnetodynamic force arising from q exerted on b:
~F[b]{*}=-k"qb~v[b]<cross>^r[b]/r{2}=(-k"/k)b~v[b]<cross>~E[b]    (7)

Magnetodynamic field arising from q in the place of b:
~B[b]{*} = ~F[b]{*}/b = -k"q~v[b]<cross>^r[b]/r{2} = 
(-k"/k)~v[b]<cross>~E[b] = (k"/k)~v[q]<cross>~E[b]                (8)

We determine "C" for the unit of electric charge and "An" for the unit of
magnetic charge. We define k=1/(4<pi><epsilon>), k'=1/(4<pi><epsilon>') and
k"=<mu>/(4<pi>). Since from the units viewpoint we have 
k: kg.m{3}.C{-2}.s{-2} , k': kg.m{3}.An{-2}.s{-2} and 
k": kg.m{2}.C{-1}.An{-1}.s{-1}, we have <epsilon>: C{2}.s{2}.kg{-1}.m{-3},
<epsilon>': An{2}.s{2}.kg{-1}.m{-3} and <mu>: kg.m{2}.C{-1}.An{-1}.s{-1}.

At present it is usual to define ~B (and in fact b) from the relation (5),
namely ~F[q]{*}=(k"/k')q~v[q]<cross>~B[q], instead of reaching the 
definition of magnetostatic field ~B in the manner presented above. In this
usual approach the coefficient k"/k' is equal to one for vacuum and is
considered without any units, ie the unit An.s.m{-1} is considered
equivalent to the unit C. So the quantity of (<mu><epsilon>){-1/2} will
have the unit of speed, and <mu><epsilon>' is equal to one for vacuum and
without any units. (This usual approach is unnecessary, I think. This 
usual definition of ~B is equivalent to say that considering the 
definition of electrostatic charge (and then knowing the quantity of k) 
and having the other quantities experimentally, we obtain the amount of
k"b from (5) ie from ~F[q]{*}=k"qb~v[q]<cross>^r[q]/r{2}, which we name
it (ie k"b) as C[1] at present. Now if we apply a pair of the same b
experimentally for the relation (3), ie ~F=k'b{2}^r/r{2}, again having
the other quantities experimentally, the amount of k'b{2} will be obtained
which we name it as C[2]. Now having k"b=C[1] & k'b{2}=C[2], if we want to 
have k"=k', we shall obtain b=C[2]/C[1] and k"=k'=C[1]{2}/C[2]. Then we define
the amount of applied b equivalent to C[2]/C[1] units of An in order that
we shall have k"/k'=1. With such a definition, after experimental measuring 
in vacuum, it is obtained that k"/k=c{-2} in which c is a quantity as big
as the speed of light in vacuum. Now having k"=k' and k"=k/c{2} we obtain
the amount of c{-2}, which is certainly very small, for the ratio of the
coefficient of the relation (8) to the coefficient of the relation (6),
ie for (k"/k)/(k"/k')=k'/k. It may be thought that in this manner the
ratio of the magnitude of the magnetodynamic field to the magnitude of the
electrostatic field is much less than the ratio of the magnitude of the
electrodynamic field to the magnitude of the magnetostatic field, or in
other words if the magnitudes of the electrostatic and magnetostatic fields
are almost the same, the magnetodynamic field will be much less than the
electrodynamic field; while this is not the case at all and the ratio of
these two coefficients, ie (k"/k)/(k"/k')=k'/k, with the supposition of
having k, depends on the k', which in turn is equal to C[2]/b{2} according
to the above explanations, and the amount of the magnitude of C[2]/b{2}
depends on the definition presented for magnetic charge (b) completely.
It is obvious that if, in this quantity, b is defined much less than 
C[2]/C[1] units of An, then the ratio k'/k will be increased very much.
In fact, what is important in the discussion of electromagnetism is that
we see from the relations (5) and (7) that the ratio of the magnitude of
~F[q]{*} to the magnitude of ~F[b]{*} is equal to one, ie the magnitudes 
of the electrodynamic and magnetodynamic forces are the same. Unfortunately
at present it is usual improperly that by using the wrong deductions and
confusing the dynamic and static fields with each other, a relation like
(F[m]/F[e] being less than or equal to (v/c)(v[1]/c) ) is concluded and 
so what is understood wrongly is that the magnetic field is much 
insignificant against the electric field (see Foundations of 
Electromagnetic Theory by Reitz, Milford and Christy, Addison-Wesley, 1979).)

Relation (8) in the form of ~B[b]{*}=-k"q~v[b]<cross>^r[b]/r{2} is in fact
the same final conclusion obtained from the experiences and experiments of
Biot, Savart and Ampere. On the basis of the above eight relations we can
present the electromagnetic theory in a simple form by using the logic of
the classical physics, as we shall do in this article. Problem, complexity
and deviation from the classical physics logic occur when we try to ignore
the existent distinction between stared forces and fields (relations (5) to 
(8)) and nonstared ones (relations (1) to (4)). For example if, as at present 
is current, we claim that ~B[q]=k'b^r[q]/r{2}, which is the magnetostatic 
field arising from the magnetic charge b in the place of the electric 
charge q, is in fact the same ~B[q]{*}=-k"q[1]~v[q]<cross>^r[q]/r{2}, which
is the magnetodynamic field arising from the electric charge q[1] in the
place of the electric charge q, with this condition that the charge q[1] is
in the same place of the charge b, and consequently we substitute ~B[q]{*}
for ~B[q] in the relation (5), ie ~F[q]{*}=(k"/k')q~v[q]<cross>~B[q], then we 
shall obtain ~F[q]{*}=k"{2}qq[1]/(k'r{2})~v[q]<cross>(~v[q[1]]<cross>^r[q]),
and on the other hand in a similar manner we shall have 
~F[q[1]]{*}=k"{2}q[1]q/(k'r{2})~v[q[1]]<cross>(~v[q]<cross>^r[q[1]]) which
is not identical with -~F[q]{*}, because considering that ^r[q[1]]=-^r[q]
it can be seen easily that the expression
~v[q[1]]<cross>(~v[q]<cross>^r[q])=~v[q]<cross>(~v[q[1]]<cross>^r[q]) is not
an identity; ie the law of action and reaction is breached easily (even
without any limit condition); besides, in principle ~B[q] and ~B[q]{*} 
cannot indicate only a single vector field, since otherwise it will be 
necessary that the vectors ^r[q] and ~v[q]<cross>^r[q] be always parallel 
with each other which is not the case obviously.

III. Static potential energy
----------------------------
As this section, please study the whole section III.A of the 6th article
of the book ("Mistakes in Electrostatics; Dreadful consequences in Modern
Physics"), and then study the following paragraph.

In a quite similar manner "the magnetostatic potential energy of an 
elective distribution of the external magnetic charge with the density
<rho>[B] is
U[B] = 1/2 <integral over V[h]>~H.~B[<rho>[B]>]dv
in which ~H is the magnetic displacement vector, and we have 
<del>.~H=<rho>[B] in which ~H=<epsilon>[0]'~B+~P[B] in which ~P[B] is an 
elective distribution of magnetostatic polarization and ~B is arising from
both ~P[B] and <rho>[B], while ~B[<rho>[B]] is the field arising only from
<rho>[B]."

IV. Modifying and completing the current relations
---------------------------------------------------
When the partial charges dv<summation over i>N[i]q[i] (in which N[i] is the
volume density of the number of the charges q[i]'s) locating in ~r[1] are
moving with respective velocities ~v[i]'s, according to the relation (8)
the partial magnetodynamic field arising from these partial charges in the
point ~r[2] is
d~B{*}(~r[2])=<mu>/(4<pi>)(1/k)<summation over i>(~v[i]<cross>~E[ir[2]])=
<mu>/(4<pi>)<summation over i>(~v[i]<cross>dv[i]N[i]q[i](~r[2]-~r[1])/|~r[
2]-~r[1]|{3})=<mu>/(4<pi>)(<summation over i>N[i]q[i]~v[i])<cross>(~r[2]-
~r[1])/|~r[2]-~r[1]|{3}dv[1]=<mu>/(4<pi>)~J(~r[1])<cross>(~r[2]-~r[1])/|
~r[2]-~r[1]|{3}dv[1],
in which ~J is the electric current density. In a quite similar manner,
by using the relation (6), we can obtain d~E{*}(~r[2]) =
-<mu>/(4<pi>)~J'(~r[1])<cross>(~r[2]-~r[[1])/|~r[2]-~r[1]|{3}dv[1], in which
~J' is the magnetic current density. Therefore:
                                                                  )
             <mu>                ~J(~r[1])<cross>(~r[2]-~r[1])    |
~B{*}(~r[2])=----<integral over V>---------------------------dv[1]|
             4<pi>                     |~r[2]-~r[1]|{3}           |
                                                                  >(9)
                                                                  |
            -<mu>               ~J'(~r[1])<cross>(~r[2]-~r[1])    |
~E{*}(~r[2])=---<integral over V>----------------------------dv[1]|
            4<pi>                      |~r[2]-~r[1]|{3}           )
                                                                        

If we operate the operator <del>[2]. on both sides of the recent relations,
we shall finally obtain

<del>.~B{*} = 0  ,  <del>.~E{*} = 0  .                           (10)

(Notice that it doesn't mean that <del>.~B=0 and <del>.~E=0.)

We have the continuity equations (<round><rho>[E]/<round>t)+<del>.~J=0 and
(<round><rho>[B]/<round>t)+<del>.~J'=0. Steady current is a current in which 
the local charge density is invariant with time, ie the charges are not
compressible and then <round><rho>/<round>t=0 and then it is necessary to 
have <del>.~J=0 and <del>.~J'=0 for steady electric and magnetic currents. If 
we operate the operator <del>[2]<cross> on both sides of the relation (9), we 
shall finally conclude that if the currents are steady, then we shall have:

<del><cross>~B{*}(~r)=<mu>~J(~r) , 
<del><cross>~E{*}(~r)=-<mu>~J'(~r)                               (11)

The curl of the vectors

~A[B](~r[2]) =                                                  )
<mu>/(4<pi>)<integral over V[1]>~J(~r[1])/|~r[2]-~r[1]|dv[1]    |
and                                                             >(12)
~A[E](~r[2]) =                                                  |
-<mu>/(4<pi>)<integral over V[1]>~J'(~r[[1])/|~r[2]-~r[1]|dv[1]  )


will be equal to ~B{*}(~r[2]) and ~E{*}(~r[2]) respectively; moreover, these
vectors have divergences and known normal components on the region 
boundaries, and then are determined uniquely. We call these unique vectors 
as magnetic vector potential and electric vector potential respectively.

If we define ~A as a vector whose components are the areas enclosed by 
projections of the supposed directed closed spatial curve C on the yz-, 
zx-, and xy-plane, then it is obvious that: <circulation over C>~r<cross>d~l
=^i<circulation over C>(ydz-zdy)+^j<circulation over C>(zdx-xdz)+^k<circula
tion over C>(xdy-ydx)=^i(A[x]+A[x])+^j(A[y]+A[y])+^k(A[z]+A[z])=2~A ==>
1/2<circulation over C>~r<cross>d~l=~A
Now if this curve is in fact a circuit carrying electric current I or
magnetic currentb I' and on definition ~m[B]=I~A and ~m[E]=-I'~A are
magnetodynamic dipole moment and electrodynamic dipole moment respectively,
then we shall have:

~m[B] = 1/2I<circulation over C>~r<cross>d~l                  )
and                                                           >  (13)
~m[E] = -1/2I'<circulation over C>~r<cross>d~l                )

By using some mathematical tricks and considering the relations (13) it
can be shown that the magnetic vector potential of a point magnetodynamic
dipole with the moment ~m[B] locating in ~r' is

~A[B](~r) = <mu>/(4<pi>)~m[B]<cross>(~r-~r')/|~r-~r'|{3},        (14)

and the electric vector potential of a point electrodynamic dipole with the
moment ~m[E] locating in ~r' is

~A[E](~r) = <mu>/(4<pi>)~m[E]<cross>(~r-~r')/|~r-~r'|{3}.        (15)

~M[B](~r')=d~m[B]/dv' and ~M[E](~r')=d~m[E]/dv' are in turn the 
magnetodynamic dipole moment density and the electrodynamic dipole moment
density (like the static cases ~P[B](~r')=d~p[B]/dv' and 
~P[E](~r')=d~p[E]/dv'). Then considering the relations (14) and (15) we can 
write

d~A[B](~r)=<mu>/(4<pi>)~M[B](~r')<cross>(~r-~r')/|~r-~r'|{3}dv' )
and                                                             >(16)
a~A[E](~r)=<mu>/(4<pi>)~M[E](~r')<cross>(~r-~r')/|~r-~r'|{3}dv' )

in which ~r' is the location of the point dipole ~M(~r')dv'.

Considering the manner presented in many of the electromagnetic textbooks
we conclude that the magnetodynamic polarization current density is
~J[M[B]]=<del><cross>~M[B], and in a quite similar manner it can be 
concluded that the electrodynamic polarization current density is
~J[M[E]]=-<del><cross>~M[E].

If we make use of <mu>[0] (related to the free space) instead of <mu> and
integrate the equations (16) over the volume of the dynamic polarized
matter, V, to obtain the proper expressions for the respective vector
potentials and take the curl of these potentials in order to obtain the
respective dynamic fields, then using some mathematical tricks we shall
finally obtain the expression

-<mu>[0]<del>1/(4<pi>)<integral 
over V>~M[B](~r').(~r-~r')/|~r-~r'|{3}dv'+<mu>[0]~M[B](~r)       (17)

for the magnetodynamic field in point ~r arising from the distribution of 
the magnetodynamic polarized matter, and the expression

-<mu>[0]<del>1/(4<pi>)<integral 
over V>~M[E](~r').(~r-~r')/|~r-~r'|{3}dv'+<mu>[0]~M[E](~r)       (18)

for the electrodynamic field in point ~r arising from the distribution of 
the electrodynamic polarized matter. (The reason for using zero subscript
is that we want to make distinct the role of the dynamic polarized matter 
from the role of the free space. So substituting <mu>[0] for <mu>, curls of
the integrals of the relations (16), which finally lead to the relations
(17) and (18), will show the contribution of the dynamic polarized matter,
without any regarding to free space, for making the dynamic field.) If we 
add the role of the nonpolarization conventional current densities, ~J and
~J', in making the dynamic fields (given by the relations (9)) to the 
expressions (17) and (18), the following general relations for the dynamic
fields, arising from the polarization and nonpolarization currents, will be
obtained:

~B{*}(~r) = <mu>[0]/(4<pi>)<integral over V>~J(~r')<cross>(~r-
~r')/|~r-~r'|{3}dv' - <mu>[0]<del>1/(4<pi>)<integral over V>~M[
B](~r').(~r-~r')/|~r-~r'|{3}dv' + <mu>[0]~M[B](~r),              (19)

~E{*}(~r) = -<mu>[0]/(4<pi>)<integral over V>~J'(~r')<cross>(~r-
~r')/|~r-~r'|{3}dv' - <mu>[0]<del>1/(4<pi>)<integral over V>~M[
E](~r').(~r-~r')/|~r-~r'|{3}dv' + <mu>[0]~M[E](~r)               (20)

As we said ~J[M[B]] and ~J[M[E]] are in turn the curls of ~M[B] and -~M[E],
and since the divergence of a curl is zero, the divergences of these 
dynamic polarization currents are zero (these currents are steady). Therefore, 
if we suppose that ~J and ~J' are the (nonpolarization) transport currents 
and in addition their divergences are zero (they are steady), then according 
to the relation (11) we shall have <del><cross>~B{*}=<mu>[0](~J+~J[M[B]])
and <del><cross>~E{*}=-<mu>[0](~J'+~J[M[E]]). Considering the curl form of
the dynamic polarization currents, we can write the recent relations in 
the following forms:

<del><cross>(1/<mu>[0]~B{*}-~M[B]) = ~J                         )
and                                                             >(21)
<del><cross>(1/<mu>[0]~E{*}-~M[E]) = -~J'                       )

We define the magnetic intensity vector, ~H{*}, and the electric intensity
vector, ~D{*}, as ~H{*}=1/<mu>[0]~B{*}-~M[B] and ~D{*}=1/<mu>[0]~E{*}-~M[E]
respectively. Then:

<del><cross>~H{*} = ~J   ,   <del><cross>~D{*} = -~J'            (22)

V. Deduction of Maxwell's equations
-----------------------------------
We now proceed to the fundamental discussions of this article more. Consider
Fig.3 which shows a piece of a U-shaped wire with a mobile wire on it, set
in a uniform magnetostatic field ~B being perpendicular to the area limited
by the wires.
                                       A
                                       |
                                       |~B               //
                               ________|______________ //____
                              / /------|-------------//-----
                          L / /        |           //
                          / /          |         //------->
                        / /                    //    ~v
                      / /___________________ //____
                     ----------------------//-----
                                X        //

     Fig.3. A loop consisting of a U-shaped wire and a mobile wire on 
            it. Magnetostatic field ~B is normal to the loop area.
              
Notice that in the following discussion there is no necessity that the 
wires to be good conductors, and they can be, for example, from wood.
Suppose that the mobile wire has a velocity ~v, as in the figure.
According to the relation (6) we know that ~E{*}=(k"/k')~v<cross>~B is
the force exerted on the electric charge unit in the mobile wire. If we
choose the unit vector ^n normal to the area surrounded by the wires by using 
the right-hand rule so that the thumb indicates the direction of ^n while
the other four fingers are in the direction of ~E{*}, then depending on the
direction of ~v we shall have:

<cap. phi>[B] = <integral over S>~B.^nda = <minus or plus>BLX ==>
d<cap. phi>[B]/dt = <minus or plus>BLdX/dt = <minus or plus>BL(<plus or 
minus>v) = -BLv or -d/dt<integral over S>~B.^nda = BvL

Now we want to find an appropriate expression for BvL. As said, we know
~E{*}=(k"/k')~v<cross>~B, ie in each point of the mobile wire the 
magnitude of ~E{*} is equal to (k"/k')vB and its direction is the same
direction of the vector ~v<cross>~B along the mobile wire. So we can write
E{*}=(k"/k')vB ==> (k'/k")E{*}L=BvL, and since E{*} is uniform in all points
of the mobile wire, we can write E{*}L=<integral over L>~E{*}.d~l, and since
E{*} is zero, in principle, in other points of the loop, we can expand the 
integral to the whole loop and write E{*}L=<circulation over C>~E{*}.d~l.
It is obvious that if the wires are conductor, they will cause produce of
a velocity for the electric charge unit in the whole loop (even in the
mobile wire itself) which , of course, this velocity produces an 
electrodynamic field normal to the path in the external magnetostatic 
field and then its respective ~E{*}.d~l is zero and we again have the same
general relation E{*}L=<circulation over C>~E{*}.d~l. Then we can write
(k'/k")<circulation over C>~E{*}.d~l=BvL and then:

-d/dt<integral over S>~B.^nda = k'/k"<ciirculation over C>~E{*}.d~l ==>
<circulation over C>~E{*}.d~l = -k"/k'd/dt<integral over S>~B.^nda

With attention to the above discussion and with notice to the right side
of the recent relation we see that the quantity being changed with time
is the area of integration not ~B, but obviously it should be presentable
that this is equivalent to this task that we assume the integration area
is a time-constant and ~B changes with time. Thus 
<circulation over C>~E{*}.d~l=-k"/k'<integral over S><round>~B/<round>t.^nda.
With some attention and sharp-sightedness we should understand that a 
mathematical careful analysis should be able to generalize the recent
equation for each position and each circuit. (Meanwhile, remembrance of this
point that the circuit may be non-conductor is recommended.) By using
Stokes' theorem the recent equation will take the form
<integral over S><del><cross>~E{*}.^nda = 
-k"/k'<integral over S><round>~B/<round>>t.^nda ,
which with its applying to the partial area da we obtain
(<del><cross>~E{*}+(k"/k')<round>~B/<round>t).^n = 0 , and since this
relation must be valid for each ^n without any attention to its direction,
<del><cross>~E{*}+(k"/k')<round>~B/<round>t=~0. (For realizing the
validity of Stokes' theorem here, we can suppose that in each point in the
space there is an infinitesimal closed circuit perpendicular to ~B in that
point, which certainly the time-changes of ~B cause the induction of some
~E{*}'s, which these ~E{*}'s will cancel each others in the common parts of 
the adjacent small circuits, or in other words, for small conductor circuits, 
the inner currents will cancel each others. Thus, in fact, in each given 
time, ~E{*} is a well-behaved point vector function in the space.)

Now we return just to the beginning of the above discussion and make some
changes in it. So, instead of notice to the relation (6) we consider the
relation (8), ie ~B{*}=-k"/k~v<cross>~E which is the force exerted on the 
magnetic charge unit in the mobile wire in Fig.4.

                                       A
                                       |
                                       |~E               //
                               ________|______________ //____
                              / /------|-------------//-----
                          L / /        |           //
                          / /          |         //------->
                        / /                    //    ~v
                      / /___________________ //____
                     ----------------------//-----
                                 X       //

     Fig.4. A loop consisting of a U-shaped wire and a mobile wire on
            it. Electrostatic field ~E is normal to the loop area.

If we choose ^n normal to the area surrounded by the wires by using the
right-hand rule so that the thumb indicates the direction of ^n while the
other four fingers are in the direction of ~B{*}, then depending on the 
direction of ~v we shall have:

<cap. phi>[E] = <integral over S>~E.^nda = <plus or minus>ELX ==>
d<cap. phi>[E]/dt = <plus or minus>ELdX/dt = <plus or minus>EL(<plus or
minus>v) = ELv or d/dt<integral over S>~E.^nda = EvL

Now we know ~B{*}=-(k"/k)~v<cross>~E, ie in each point in the mobile wire
the magnitude of ~B{*} is equal to (k"/k)vE and its direction is the same 
direction of the vector ~E<cross>~v along the mobile wire. Thus
B{*}=(k"/k)vE ==> (k/k")B{*}L=EvL, and since with a discussion similar
to the previous discussion we can write B{*}L=<circulation over C>~B{*}.d~l,
we have EvL=(k/k")<circulation over C>~B{*}.d~l, and then

d/dt<integral over S>~E.^nda=(k/k")<circulation over C>~B{*}.d~l ==>
<circulation over C>~B{*}.d~l=(k"/k)<integral over S><round>~E/<round>t.^nda,

and with a discussion similar to the previous one, we shall obtain
<del><cross>~B{*}-(k"/k)<round>~E/<round>t=~0. (Notice that existence of
magnetic current is not necessary, because as we said the circuit could be
non-conductor.)

We said before that if the currents were steady, ie the charge densities
were not being changed with time, we would have the relations (11), and
now we have shown in fact that under the same condition, ie if the currents 
are steady, we have <del><cross>~B{*}=(k"/k)<round>~E/<round>t and 
<del><cross>~E{*}=-(k"/k')<round>~B/<round>t. This condition is implied by 
perceiving this fact that as the charge density hasn't been changed during 
the displacement of the mobile wire, the charge density won't also be
changed in the position of the point loop, which is in fact a point of the
line of the current being in fact the same movement of the sources, during
the displacement of the sources causing the time variation of the static
fields. (The movement of the sources is in fact the same currents in the
relations (9) which ultimately lead to the relations (11).) Thus we have

~J'=<epsilon>'<round>~B/<round>t,~J=<epsilon><round>~E/<round>t, (23)

and in any case:

<del><cross>~B{*} = <mu><epsilon><round>~E/<round>t             )
and                                                             >(24)
<del><cross>~E{*} = -<mu><epsilon>'<round>~B/<round>t           )

Now again we return to the beginning of the previous discussions related 
to the loop circuit. Depending on the kind of material used for the wires
(including the mobile wire) the force exerted on the unit charge, ie ~E{*},
can move the charges in the circuits and cause them to reach the final 
state of equilibrium in motion very soon and thus produce an electric 
current. The work which ~E{*} does in transferring the unit electric charge
from one end of the mobile wire to its other end is E{*}L or (k"/k')BvL.
We can attribute this work, <cursive E>, to some potential difference
formed between the two ends of the mobile wire, <cursive V>. Consequently
we have

(<circulation over C>~E{*}.d~l=The work done on the unit electric charge 
in a complete cycle)=(k"/k')BvL & (k'/k")<circulation over C>~E{*}.d~l=
(-d/dt<integral over S>~B.^nda=-d<cap. phi>[B]/dt) ==>
<cursive V>=<cursive E>=-(k"/k')d<cap. phi>[B]/dt,

and then, considering the circulation direction on the loop for showing ^n,
which is in agreement with the right-hand rule mentioned at the beginning
of this section, Lenz's law is obtained which states: "If, passing the time,
<cap. phi>[B] is increased in the circuit, <cursive E> has such form that
as a result of the electric current arising from it, some magnetodynamic
flux is produced that wants to decrease <cap. phi>[B]. And if, passing the
time, <cap. phi>[B] is decreased in the circuit, <cursive E> has such form 
that as a result of the electric current arising from it, some magnetodynamic
flux is produced that wants to increase <cap. phi>[B]."

And referring to the next loop discussion we see that the work done by
~B{*} in transferring the unit magnetic charge from one end to the other
end of the mobile wire is B{*}L or (k"/k)EvL, which we show it by 
<cursive B>, and we can attribute it to a potential difference, <cursive V>',
formed between the two ends of the mobile wire. Consequently we have

(<circulation over C>~B{*}.d~l=The work done on the unit magnetic charge
in a complete cycle)=(k"/k)EvL & (k/k")<circulation over C>~B{*}.d~l=
(d/dt<integral over S>~E.^nda=d<cap. phi>[E]/dt) ==>
<cursive V>'=<cursive B>=(k"/k)d<cap. phi>[E]/dt.

Here with attention to this fact that the electrodynamic field arising from
a magnetic current is obtained by using the left-hand rule, no right-hand
rule, (so that the thumb of the left hand indicates the direction of the
magnetic current while its other four fingers are in the direction of the
electrodynamic field; pay attention to the relation (8) in comparison with
the relation (6)) and considering the mentioned circulation direction on the
loop for showing ^n, we have Lenz's law again which states: "If, passing
the time, <cap. phi>[E] is increased in the circuit, <cursive B> has such 
form that as a result of the magnetic current arising from it, some
electrodynamic flux is produced that wants to decrease <cap. phi>[E]. And
if, passing the time, <cap. phi>[E] is decreased in the circuit, 
<cursive B> has such form that as a result of the magnetic current arising
from it, some electrodynamic flux is produced that wants to increase
<cap. phi>[E]."

The above two results can also be shown as:

<cursive E> = -<mu><epsilon>'d<cap. phi>[B]/dt                  )
and                                                             >(25)
<cursive B> = <mu><epsilon>d<cap. phi>[E]/dt                    )

With attention to the relation (9) and by using the identity ~Jdv=Id~l,
in the case of a rigid and motionless isolated circuit we have:

~B{*}(~r[2]) = <mu>/(4<pi>)I[1]<integral over 1>(d~l[1]<cross>(~r[2]-
~r[1])/|~r[2]-~r[1]|{3}) & (The circuit is rigid and motionless.) ==>
d~B{*}(~r[2])/dI[1] = ~B{*}(~r[2])/I[1],                         (26)
<cap. phi>[B]{*} = <integral over S>~B{*}(~r[2]).^nda ==>
d<cap. phi>[B]{*}/dI[1]=<integral over S>d~B{*}(~r[2])/dI[1].^nda,(27)
(26) & (27) & (The circuit is rigid and motionless.) ==>
d<cap. phi>[B]{*}/dI[1] = <cap. phi>[B]{*}/I[1],                  (28)
(The circuit is rigid and motionless.) & <cap. phi>[B]{*} = <integral
over S>~B{*}(~r[2]).^nda ==> d<cap. phi>[B]{*}/dt = <integral over
S>d~B{*}(~r[2])/dt.^nda = <integral over S>d~B{*}(~r[2])/dI[1] dI[
1]/dt.^nda = d<cap. phi>[B]{*}/dI[1] dI[1]/dt    )
                                                 > ==>
                                         (28)    )
d<cap. phi>[B]{*}/dt = <cap. phi>[B]{*}/I[1] dI[1]/dt             (29)

In a similar manner we have 
d<cap. phi>[E]{*}/dt = (d<cap. phi>[E]{*}/dI[1]')(dI[1]'/dt) or 
d<cap. phi>[E]{*}/dt = (<cap. phi>[E]{*}/I[1]')(dI[1]'/dt). We define
L[B]{*}=d<cap. phi>[B]{*}/dI and L[E]{*}=d<cap. phi>[E]{*}/dI' as 
magnetodynamic and electrodynamic inductances respectively. The 
magnetodynamic inductance, L[B]{*}, is in fact not useful so much, and its
chief rolr is in fact giving order to the magnetostatic dipoles of the
medium and so producing the magnetostatic inductance L[B]=d<cap. phi>[B]/dI.
Probably in many cases we have <cap. phi>[B]=a'<cap. phi>[B]{*} practically
which itself is arising from ~B=a'~B{*} which we prescribe this as the 
definition of the medium being "magnetolinear" in which a' is the proportion 
constant which is nonnegative. By presenting similar matters we can attain 
to the relation ~E=a~E{*} which we prescribe it as the definition of the
medium being "electrolinear" in which a is the proportion constant that 
is nonnegative. Trying to justify these proportions we suppose (and probably
posterity will show) that the space of any substance is full of some much
tiny electrostatic and magnetostatic dipoles that have almost random
orientations in general state. By exerting a dynamic field in this space 
some of the relevant dipoles which have weaker bonds will be put into 
order in the direction of the field, and it is natural that they will
produce a relevant static field in the same direction of the dynamic field
in a point of the space between the dipoles. It is obvious that if for
instance the exerted dynamic field becomes twofold, in addition to the
oriented previous group of the relevant static dipoles another equinumber
group of these dipoles, which of course have stronger bonds, will be put
into order in the direction of the field, and therefore the static field
will become twofold in the same point between the dipoles. In this same
manner the proportion of the exerted dynamic field to the static field 
produced in the space between the dipoles as a result of the orientation 
of the dipoles in the direction of the dynamic field will become meaningful.
We should know that these tiny dipoles don't belong to the substance in
question, but have penetrated it.

We suppose that <cap. phi>[B] to be a point function of I (what that I 
don't think that is valid, for instance, for the ferromagnetic materials).
Therefore, d<cap. phi>[B]/dI=a'd<cap. phi>[B]{*}/dI or L[B]=a'L[B]{*}. (We 
should know that we can say d<cap. phi>[B]/dI=<cap. phi>[B]/I still, because 
<cap. phi>[B]=a'<cap. phi>[B]{*} and d<cap. phi>[B]{*}/dI=<cap. phi>[B]{*}/I,
and I think that the necessity of distinguishing between <cap. phi>[B]/I
and d<cap. phi>/dI which is necessary sometimes is because of the deviation
from linearity of substances (pay attention to the definition of linearity
presented above).) Thus

d<cap. phi>[B]{*}/dt = d<cap. phi>[B]{*}/dI dI/dt = L[B]{*}dI/dt =
L[B]/a' dI/dt & <cursive E> = -<mu><epsilon>'d<cap. phi>[B]/dt =
-<mu><epsilon>'d<cap. phi>[B]/dI dI/dt == -<mu><epsilon>'L[B]dI/dt ==>
<cursive E> = -<mu><epsilon>'a'L[B]{*}dI/dt.

(The relation that is under consideration instead of this relation in the
present electromagnetic books has the form <cursive E>=-LdI/dt in which 
L=d<cap. phi>/dI, and of course the magnetodynamic flux is in mind from 
<cap. phi>, without any coefficient like a' that can show the intervention
of the role of the medium in <cursive E>. Now since <cursive E> is 
measurable practically, if we make the conditions so that all the effective 
parameters remain unaltered except that the mediums of experiment are made
altered in the magnetic respect, then if <cursive E> will be altered with
altering the mediums, the aspect presented in this article will be
confirmed practically.) In a quite similar manner we obtain L[E]=aL[E]{*}
and <cursive B>=<mu><epsilon>aL[E]{*}dI'/dt for an electrolinear medium.

So far only isolated circuits have been under study, so that the totality 
of the flux passing through the circuit was due to the current of the
circuit itself. We can eliminate this limitation by supposing that there
exist n circuits with respective numbers 1, 2, 3, ..., n. In this state,
if we suppose that the medium is full-linear (ie both magnetolinear and
electrolinear), then we shall have

<cursive E>[i]=-<mu><epsilon>'<summation from j=1 to n>d<cap. phi>[Bij]/dt=
-<mu><epsilon>'<summation from j=1 to n>>d<cap. phi>[Bij]/dI[j]dI[j]/dt=
-<mu><epsilon>'<summation from j=1 to n>>M[Bij]dI[j]/dt
and
<cursive B>[i]=<mu><epsilon><summation from j=1 to n>d<cap. phi>[Eij]/dt=  
<mu><epsilon><summation from j=1 to n>d<cap. phi>[Eij]/dI[j]'dI[j]'/dt=
<mu><epsilon><summation from j=1 to n>M[Eij]dI[j]'/dt.

It is obvious that M[Eii]=L[Ei] and M[Bii]=L[Bi]. We know that M[Bij]=
d<cap. phi>[Bij]/dI[j]=a'd<cap. phi>[Bij]{*}/dI[j]=a'<cap. phi>[Bij]{*}/I[j]
and since M[Bij]{*}=d<cap. phi>[Bij]{*}/dI[j], we have M[Bij]{*}=
<cap. phi>[Bij]{*}/I[j] and also M[Bij]=a'M[Bij]{*}; and in a similar manner
M[Eij]{*}=<cap. phi>[Eij]{*}/I[j]' and also M[Eij]=aM[Eij]{*}. Now suppose
that i=2 and j=1, then <cap. phi>[B21]{*}=<integral over S2>~B[21]{*}.^nda[2]
=<integral over S2>(<mu>/(4<pi>)I[1]<circulation over C1>(d~l[1]<cross>(~r[2]
-~r[1])/|~r[2]-~r[1]|{3})).^nda[2]=<mu>((4<pi>)I[1]<integral over S2>(<circu
lation over C1>(d~l[1]<cross>(~r[2]-~r[1])/|~r[2]-~r[1]|{3})).^nda[2]; and
by using the mathematical relation <circulation over C1>(d~l[1]<cross>(~r[2]-
~r[1])/|~r[2]-~r[1]|{3})=<del>[2]<cross><circulation over C1>(d~l[1]/|~r[2]-
~r[1]|) we have the relation M[B21]{*}=<cap. phi>[B21]{*}/I[1]=<mu>/(4<pi>)<
integral over S2><del>[2]<cross>(<circulation over C1>(d~l[1]/|~r[2]-~r[
1]|)).^nda[2] and so, by using Stokes' theorem, we have M[B21]{*}=<mu>/(4<
pi>)<circulation over C2><circulation over C1>(d~l[1].d~l[2]/|~r[2]-~r[1]|),
which is in fact the same Neumann's formula for the mutual inductance. In
a similar manner we shall have M[E21]{*}=-<mu>/(4<pi>)<circulation over 
C2><circulation over C1>(d~l[1].d~l[2]/|~r[2]-~r[1]|). (d~l[1] and d~l[2]
are in the directions of the respective currents, and as it is seen we have
M[Bij]{*}=M[Bji]{*} and M[Eij]{*}=M[ji]{*}.)

We want to see in a simple manner how an electromagnetic wave propagates
through the space carrying the wave with attention to the supposition of the 
existence of the much tiny static dipoles filling up this supposed 
full-linear space. This matter can be perceived by observing Fig.5 in which 
it is supposed that the + enclosed with the square is the representative of 
the accelerated electric charge or in fact the oscillator producing the 
electromagnetic wave.

 A ._                A   ._                A   ._                A
 | |\  ___           |   |\  ___           |   |\  ___           |
 |   /'   `\    ___  |     /'   `\    ___  |     /'   `\    ___  |
,|, (  N    ) /'   `\|_   (  N    ) /'   `\|_   (  N    ) /'   `\|_
|+|  \_   _/ (  - /'   `\  \_   _/ (  - /'   `\  \_   _/ (  - /'   `\
^^^  /'^^^`\  \_ (  +    ) /'^^^`\  \_ (  +    ) /'^^^`\  \_ (  +    )
    (  S\   )   ^|\_   _/ (  S\   )   ^|\_   _  (  S\   )   ^|\_   _/
     \_  `\/     |  ^^^    \_  `\/     |  ^^^    \_  `\/     |  ^^^
       ^^^ `\!   |           ^^^ `\!   |           ^^^ `\!   |
            ^`   V                ^`   V                ^`   V

    Fig.5. How an electromagnetic wave is produced and propagated.

(Use properly the right and left hand rules in each point for finding the
next movements.) Meanwhile we see easily why ~E{*} and ~B{*} are 
perpendicular to each other. This aspect shows that how beautiful the 
electric and magnetic fields are complementaries to each other or in fact 
producers of each other, not as two elements as if being quite distinct
from each other that have been set by the side of each other by chance,
ie the aspect pursued in the present electromagnetic texts! It also shows
a produced current due to the turn or in fact orientation of the static 
dipoles. With attention to the proportion of the static field to the dynamic
one in the above supposed full-linear medium, these currents are obviously
unidirectional with the fields generating them (compare with the relations
(23)). It is obvious that the time changes of the wave, for instance the
sinusoidal changes of the wave, are related to the changes of the source
movement. More explanations about this aspect will be presented in the next
section. The confirmation for this aspect and, in principle, the confirmation
of this matter that the space through which the wave passes should contain 
some static charges as wave carrier is that like the other cases in physics
that we reach to a wave equation, the equations by which we reach to the wave 
equation should be valid for each section of the carriers making the wave
path, and this matter requires existence of some static charges in the
whole wave path in order that the fundamental equations can be executed for
them. We now proceed to these fundamental equations. We have the following
equations instead of Maxwell's equations with this supposition that the
electromagnetic wave carrier mediums are full-linear and considering the
discussions presented so far.

<del>.~B{*} = 0                                                 )
<del>.~E{*} = 0                                                 |
                                                                >(30)
<del><cross>~B{*} = <mu><epsilon>a<round>~E{*}/<round>t         |
<del><cross>~E{*} = -<mu><epsilon>'a'<round>~B{*}/<round>t      )

It may be said that while considering the discussions about the static fields
we know that the curl of the static fields must be zero, considering the
proportion of the static to the dynamic field in a full-linear medium we can
deduce from the two last equations of (30) that the curl of the static fields 
of one kind is proportional to the time derivative of the static fields of 
the other kind or in other words to the respective generated currents (see
(23)). The answer to this problem should be searched in the potential 
discontinuity in a continuous distribution of the static point dipoles (eg
it is clear that in passing through a dipole layer, the potential finds a 
jump, which is in fact the same potential changing in passing across the
infinitesimal thickness of the layer (see Classical Electrodynamics by
Jackson, John Wiley & Sons, 1962)). After accepting the existence of such
a discontinuity, it is natural that we should also accept that the curl of
the static field is no longer identical with zero in each point of this 
distribution, because otherwise the existence of a continuous potential
would be necessary.

By using the relation ~B{*}(~r[2]) =
<mu>/(4<pi>)<integral over L>(Id~l<cross>(~r[2]-~r[1])/|~r[2]-~r[1]|{3}),
which is the same relation of (9) in which we have made use of the relation
~Jdv=Id~l for a wire carrying electric current I, the magnetodynamic field
arising from the electric current I in a long straight wire, lying along
the x-axis from minus infinity to plus infinity, will be 
~B{*}=<mu>I^k/(2<pi>a) in a distance "a" from the wire on the y-axis.
Now we proceed to another different problem. We consider an infinite long
wire along the x-axis which carries the current I from minus infinity to
plus infinity. We suppose that there is a uniform magnetostatic field ~B
along the whole length of the wire. We proceed to accounting the 
electrodynamic force exerted on the unit length of the wire by using
the relation (5):

d~F[dq]{*} = (k"/k')dq~v[dq]<cross>~B = (k"/k')dq(d~l/dt)<cross>~B
= (k"/k')Id~l<cross>~B  ==> (d~F[dq]{*}/dl) = (k"/k')I^i<cross>~B

This means that if, for example, ~B is in the positive or negative direction
of the z-axis, the force will be in the negative or positive direction of
the y-axis respectively. Another usual mistake is replacing ~B{*}, arising
from another carrying current wire parallel to this wire, instead of ~B
in the recent relation and, with this contrivance, justifying the 
attractive force between two parallel wires with same directions for 
currents and the repulsive force between them with different directions 
for their currents; while we should say that as we said the magnetodynamic
field ~B{*} causes production of the magnetostatic field ~B in the
magnetolinear medium around the wire, ie we have ~B=a'~B{*}. And then we
should substitute a'~B{*} for ~B in the recent formula. It is
probable that this feature be correct, but even if so, its role
in the force between two currents is so much smaller than the
real cause of the force between two currents which is the subject
of one of my papers.              

VI. Wave equations and Poynting vectors
---------------------------------------
By using the equations (30) we have:

(<del><cross><del><cross>~B{*} = -<del>{2}~B{*}) = 
(<mu><epsilon>a<round>/<round>t(<del><cross>~E{*}) = 
<mu><epsilon>a<round>/<round>t(-<mu><epsilon>'a'<round>~B{*}/<round>t) = 
-<mu>{2}<epsilon><epsilon>'aa'<round>{2}~B{*}/<round>t{2}) ==>          
<del>{2}~B{*}
= <mu>{2}<epsilon><epsilon>'aa'<round>{2}~B{*}/<round>t{2}       (31)
and
(<del><cross><del><cross>~E{*} = -<del>{2}~E{*}) =
(-<mu><epsilon>'a'<round>/<round>t(<del><cross>~B{*}) =
-<mu><epsilon>'a'<round>/<round>t(<mu><epsilon>a<round>~E{*}/<round>t)=
-<mu>{2}<epsilon><epsilon>'aa'<round>{2}~E{*}/<round>t{2}) ==>
<del>{2}~E{*}
= <mu>{2}<epsilon><epsilon>'aa'<round>{2}~E{*}/<round>t{2}       (32)

The equations (31) and (32) are wave equations indicating electromagnetic 
wave propagating with the speed v=(<mu>{2}<epsilon><epsilon>'aa'){-1/2} in 
the full-linear medium of a and a'.

We have these cases:
With attention to Lenz's law and Kirchhoff's loop law, immediately after
closing the circuit in Fig.6(a) we have <cursive V>-(-<cursive E>)-IR=0 ==>
<cursive V>=IR-<cursive E>; and immediately after opening the circuit in
Fig.7(a) we have <cursive V>+<cursive E>-IR=0 ==> 
<cursive V>=IR-<cursive E>.

    ______________                     ______________
   |              |                   |              |
   \              $                   \              $
   /              $ *                 /              $ *
   \ R            $ | -<cursive E>    \ R'           $ | -<cursive B>
   /       |      $ V                 /       |      $ V
   |______||______|                   |______||______|
          ||                                 ||
           |                                  |
         *---> <cursive V>                  *---> <cursive V>'

          (a)                                (b)

     Fig.6. States of electric (a) and magnetic (b) closed circuits 
            immediately after closing then. In these figures 
            <cursive E> and <cursive B> are negative.

    ______________                     ______________
   |              |                   |              |
   \              $ A                 \              $ A
   /              $ |                 /              $ |
   \ R            $ * <cursive E>     \ R'           $ * <cursive B>
   /       |      $                   /       |      $
   |______||______|                   |______||______|
          ||                                 ||
           |                                  |
         *---> <cursive V>                  *---> <cursive V>'

          (a)                                (b)

     Fig.7. States of electric (a) and magnetic (b) closed circuits
            immediately after opening them. In these figures
            <cursive E> and <cursive B> are positive.

Similarly with attention to Lenz's law and Kirchhoff's loop law, immediately
after closing the circiuit in Fig.6(b) we have
<cursive V>'-(-<cursive B>)-I'R'=0 ==> <cursive V>'=I'R'-<cursive B>; and
immediately after opening the circuit in Fig.7(b) we have 
<cursive V>'+<cursive B>-I'R'=0 ==> <cursive V>'=I'R'-<cursive B>. Thus if a 
source <cursive V> is exerted to a circuit, we have <cursive V>=IR-<cursive E> 
in which <cursive E> is the induced electromotive force, and if a source
<cursive V>' is exerted to a circuit, we have <cursive V>'=I'R'-<cursive B>
in which <cursive B> is the induced magnetomotive force.

In the following discussion we suppose that none of the circuits are in any
external field. Now we have:

<cursive V>=IR-<cursive E> & dq=Idt & <cursive E>
= -<mu><epsilon>'d<cap. phi>[B]/dt
==> <cursive V>dq=<mu><epsilon>'Id<cap. phi>[B]+I{2}Rdt         (33)
and
<cursive V>'=I'R'-<cursive B> & db=I'dt & <cursive B>
= <mu><epsilon>d<cap. phi>[E]/dt
==> <cursive V>'db=-<mu><epsilon>I'd<cap. phi>[E]+I'{2}R'dt     (34)

The left side terms in the relations (33) and (34) are the work done on the
partial charge, and the second terms on the right are that part of this work
which is wasted in the circuit in the form of heat, and the first terms on
the right are the work done in the circuit for opposing the induced motive
forces and in fact in the case of rigidity and immobility of the circuit 
this work will be spent totally for making the field around the circuit and
naturally will be stored in it as the potential energy. For better 
understanding of this matter we should say that it is only after ordering 
the tiny static dipoles of the medium by the dynamic field arising from
the current, and so producing the static field, that the recent work will be 
stored in this static field as the potential energy (because in the 
expressions <cursive E>=-<mu><epsilon>'d<cap. phi>[B]/dt and 
<cursive B>=<mu><epsilon>d<cap. phi>[E]/dt we see the static fluxes, not the 
dynamic ones). Of course attention to this point is important that this
potential energy is only that portion of the static potential energy 
presented in the section III that easily, ie without any necessity to
changing the structural composition of the mentioned dipoles, is changeable 
to other forms of energy particularly heat (notice that if it is 
supposed that the static potential energy of a nonzero charge to be
changeable to heat (or other forms of energy) totally, it will be required
that all the differential parts of the charge to disintegrate from one
another and then the structural composition of the charge to be changed).
Suppose that our system is full-linear and includes n stationary circuits.
For accounting the magnetic or electric potential energy (mentioned above) 
arising from the currents of the system, it is sufficient to integrate the
expressions <mu><epsilon>'<summation from i=1 to n>I[i]d<cap. phi>[Bi] and
-<mu><epsilon><summation from i=1 to n>I[i]'d<cap. phi>[Ei] from the zero
flux situation to the final values of the fluxes. (In these two expressions
<cap. phi>[Bi] and <cap. phi>[Ei] are arising from all the currents, not 
only arising from the current in the circuit i.) Since this energy is 
independent of the way in which the currents are brought to their final set
of values, we choose a way in which at any instant of time all the currents
will be at the same fraction of their final values. We call this fraction as
<alpha>. Now in the first degree, before all, we have:

<cap. phi>[Bi]{*}=<integral over Si>~B[i]{*}.^nda ==>
d<cap. phi>[Bi]{*}/dI[j]=<integral over Si>d~B[i]{*}/dI[j].^nda (35')
~B[i]{*}=<summation from j=1 to n>~B[ij]{*} ==> 
d~B[i]{*}/dI[j]=d~B[ij]{*}/dI[j] )
                                 > ==> d~B[i]{*}/dI[j]=1
 d~B[ij]{*}/dI[j]=~B[ij]{*}/I[j] )             =~B[ij]{*}/I[j]  (35")
(35')&(35")==>d<cap. phi>[Bi]{*}/dI[j]=<cap. phi>[Bij]{*}/I[j]   (35)

(we are attentive that each of the expressions of both sides of the equality
(35) is related only to the structural shape of the circuit and that it is
unimportant that ~B[ij]{*} and I[j] belong to which instant of time and then 
we suppose that in the right side of this relation I[j] and ~B[ij]{*} are
the final values of these quantities.)

Now we show the electric and magnetic currents which have not reached to 
their final values and are related to the circuit j as (I[j])' and (I[j]')'
respectively. We have:

d<cap. phi>[Bi]{*}/d(I[j])'=<cap. phi>[Bij]{*}/I[j] ==> d<cap. phi>[Bi]{*}=<
cap. phi>[Bij]{*}d(I[j])'/I[j]        )
<cap. phi>[Bij]{*}=<cap. phi>[Bij]/a' >==>d<cap. phi>[Bi]=<cap. phi>[
<cap. phi>[Bi]{*}=<cap. phi>[Bi]/a'   )
Bij]d(I[j])'/I[j]                                                    )
                                                                     >
(I[j])'=<alpha>I[j]==>d(I[j])'=I[j]d<alpha>==>d(I[j])'/I[j]=d<alpha> )
==> d<cap. phi>[Bi]=<cap. phi>[Bij]d<alpha>                      (36)

Now we know that d<cap. phi>[Bi]=<summation from j=1 to n><round><cap. 
phi>[Bi]/<round>(I[j])'d(I[j])' and the expression (36) is in fact the same
<round><cap. phi>[Bi] in the recent relation and so d<cap. phi>[Bi]=
<summation from j=1 to n><cap. phi>[Bij]<round><alpha>/<round>(I[j])'d(I[j])'
and since <alpha> is related to each (I[j])' in the one-variable form of
<alpha>=(I[j])'/I[j], the sign <round> will become changed into d and
we shall have d<cap. phi>[Bi]=<summation from j=1 to n><cap. phi>[Bij]d<al
pha>=<cap. phi>[Bi]d<alpha> in which <cap. phi>[Bi] is arising from all the
currents. In a similar manner we have d<cap. phi>[Ei]=<cap. phi>[Ei]d<alpha>.
Thus <integral from <alpha>=0 to 1><mu><epsilon>'<summation from i=1 to
n><alpha>I[i]<cap. phi>[Bi]d<alpha>=<mu><epsilon>'<summation from i=1 to 
n>I[i]<cap. phi>[Bi]<integral from <alpha>=0 to 1><alpha>d<alpha>=
<mu><epsilon>'/2<summation from i=1 to n>I[i]<cap. phi>[Bi]=U[B]{*} and
<integral from <alpha>=0 to 1>-<mu><epsilon><summation from i=1 to n><al
pha>I[i]'<cap. phi>[Ei]d<alpha>=-<mu><epsilon><summation from i=1 to 
n>I[i]'<cap. phi>[Ei]<integral from <alpha>=0 to 1><alpha>d<alpha>=
-<mu><epsilon>/2<summation from i=1 to n>I[i]'<cap. phi>[Ei]=U[E]{*}. We 
know ~B{*}=<del><cross>~A[B] and ~E{*}=<del><cross>~A[E] and <cap. phi>[Bi]=
<integral over Si>~B[i].^nda=<integral over Si>a'~B{*}.^nda=a'<integral over 
Si><del><cross>~A[Bi].^nda=a'<circulation over Ci>~A[Bi].d~l[i] and <cap. 
phi>[Ei]=<integral over Si>~E[i].^nda=<integral over Si>a~E{*}.^nda= a<integ
ral over Si><del><cross>~A[Ei].^nda=a<circulation over Ci>~A[Ei].d~l[i].
(We have attention that the direction of ^n is determined by considering 
the direction of the current in the circuit i and by using the right-hand
rule, and so the circulation is also in the direction of the current.) 
And so U[B]{*}=1/2<mu><epsilon>'a'<summation over i><circulation over
Ci>I[i]~A[Bi].d~l[i] and U[E]{*}=-1/2<mu><epsilon>a<summation over i><cir
culation over Ci>I[i]'~A[Ei].d~l[i]. In a general state each "circuit" is
a closed path in a given linear medium which traces a current density line.
So we can set ~Jdv instead of I[i]d~l[i] and ~J'dv instead of I[i]'d~l[i]
and <integral over V> instead of <summation over i><circulation over Ci>.
So we have U[B]{*}=1/2<mu><epsilon>'a'<integral over V>~J.~A[B]dv and
U[E]{*}=-1/2<mu><epsilon>a<integral over V>~J'.~A[E]dv. Now using the 
vector identity <del>.(~F<cross>~G)=(<del><cross>~F).~G-~F.(<del><cross>~G)
and using the equations (11) we obtain U[B]{*}=1/2<epsilon>'a'<integral
over V>(<del><cross>~B{*}).~A[B]dv=1/2<epsilon>'a'(<integral over V>~B{*}.(
<del><cross>~A[B])dv-<circulation over S>(~A[B]<cross>~B{*}).^nda) and
U[E]{*}=1/2<epsilon>a<integral over V>(<del><cross>~E{*}).~A[E]dv=1/2<epsi
lon>a(<integral over V>~E{*}.(<del><cross>~A[E])dv-<circulation over S>(
~A[E]<cross>~E{*}).^nda). We take the volume V and the closed surface S
infinitely large (an infinite sphere about the center of which our system
has been set). Now we know that ~B{*} and ~E{*} fall off as fast as 
r{-2} (r is the radius of this sphere) and ~A[B] and ~A[E] fall off as fast
as r{-1} and S is proportional to r{2}. Thus the second integrals, surface 
integrals, in both the recent relations fall off as fast as
r{-2}r{-1}r{2}=r{-1} so that when the volume and surface go to infinity
these integrals will become zero, and considering that <del><cross>~A[B]=
~B{*}=~B/a' and <del><cross>~A[E]=~E{*}=~E/a we obtain

U[B]{*} = 1/2<epsilon>'a'<integral over V[h]>~B{*}.B/a'dv       )
= 1/2<epsilon>'<integral over V[h]>~B{*}.~Bdv                   |
and                                                             >(37)
U[E]{*} = 1/2<epsilon>a<integral over V[h]>~E{*}.~E/adv         |
= 1/2<epsilon><integral over V[h]>~E{*}.~Edv                    )

in which V[h] is the whole space. As we said before, the dynamic potential
energies, U[B]{*} and U[E}{*}, are not separate from the static potential
energies, U[B] and U[E], but they are only those parts of the static 
potential energies which are related to the energy spent for orienting the
tiny static dipoles and do not include the energy needed for gathering 
the partial ingredient parts of the dipoles themselves from infinity
differential by differential. These dynamic energies are easily changeable
to other forms of energy especially heat.
Now we return to the equations (30). We have

<del><cross>~B{*} = <mu><epsilon><round>~E/<round>t ==>
~E{*}.<del><cross>~B{*} = <mu><epsilon>~E{*}.<round>~E/<round>t
= 1/2<mu><epsilon><round>/<round>t(~E{*}.~E)                   (38')
<del><cross>~E{*} = -<mu><epsilon>'<round>~B/<round>t ==>
~B{*}.<del><cross>~E{*} = -<mu><epsilon>'~B{*}.<round>~B/<round>t
= -1/2<mu><epsilon>'<round>/<round>t(~B{*}.~B)                 (38")
<del>.(~E{*}<cross>~B{*})
=~B{*}.<del><cross>~E{*}-E{*}.<del><cross>B{*} )
                                         (38') > ==>
                                         (38") )
<del><cross>(1/<mu>~E{*}<cross>~B{*})
=-<round>/<round>t(1/2<epsilon>'~B{*}.~B + 1/2<epsilon>'~E{*}.~E) (38)

With attention to the equations (37) we show the right side of the relation
(38) by -<round>u/<round>t in which u is the energy density consisting of
sum of the densities of the electrodynamic and magnetodynamic potential 
energies (which are changeable to heat). The left side of this relation
can be written as <del>.~S in which ~S is the same 1/<mu>~E{*}<cross>~B{*}
and we call it as Poynting vector. Thus

<del>.~S + <round>u/<round>t = 0 .                              (39)

This equation is a continuity equation and states that if the energy
density is decreased (or increased) in a point, then necessarily some energy
has gone out of this point (or some energy has entered into this point).
Therefore, ~S is the local current of energy per unit time per unit area
(as unitary investigation of ~S shows this matter too).

What we deduce from the above discussion is that ~S=1/<mu>~E{*}<crss>~B{*}
also shows the direction of propagation of the wave, the wave that its 
energy is easily changeable to other forms of energy especially heat. Now we
want as far as possible to present a simple physical interpretation about
this matter that ~E{*}<cross>~B{*} shows the direction of propagation of the
electromagnetic wave considering the tiny particle (or in fact dipole) model
about which we have explained before. For simplicity consider the 
orientational movement of only the positive charges (ie N-poles and positive
electric charges) of the dipoles. Suppose that some electric current is 
flowing in the positive direction of a straight wire set along the x-axis
as in Fig.8.
                    | |
     A              |A|
   x |              |||         /^\             /^\             /^\
     |              |||     _  /   \        _  /   \        _  /   \
     |     z        | |_,-'` |/  B  \  _,-'` |/  B' \  _,-'` |/     \
     *------>+ A[1] _*'   A  +       +'  A'  +       +'      +       +
 y /`        |  _-'`| |
!/`           '`    | |
`^                  | |
                    | |

    Fig.8. Why ~E{*}<corss>~B{*} shows the direction of propagation
           of the electromagnetic wave.

This current causes the moving (or in fact the orientation) of the positive
magnetic charges of the magnetostatic tiny dipoles of the medium in a range
the extent of which is determined by the physical conditions of the medium,
indicated here by A, toward the negative direction of the y-axis, according
to the right-hand rule. This movement causes the moving of the positive
electric charges of the electrostatic tiny dipoles of the medium in a range
which is again determined by the same physical conditions, indicated here by
B, toward the positive direction of the x-axis, according to the left-hand 
rule. This recent movement of the electric particles causes not only the
returning of A to its initial equilibrium state but also the moving of the 
positive magnetic particles in the range A' toward the negative direction
of the y-axis, and this recent movement of the magnetic particles, in turn,
not only restores B to its initial equilibrium state but also causes the
moving of the positive electric particles in the range B', and in this 
manner the wave movement will be transferred. Now we may encounter with 
this question that whether this wave movement doesn't have continuity as 
in Fig.8. The answer to this question is that this is only one half of the
work and the other half is that the current in the wire also causes the 
moving of the magnetic particles of the medium in the range A[1] according
to the right-hand rule, and this movement, which is not restored to its
initial equilibrium state as a result of a similar agent, causes the 
moving of the positive electric particles in the range A in the negative
direction of the x-axis according to the left-hand rule, and this recent 
movement, in turn, not only restores A[1] to its initial equilibrium state
but also causes the moving of the positive magnetic particles in the range
B toward the positive direction of the y-axis, both according to the 
right-hand rule. Again in this manner the wave movement will be transferred.
Consequently on the whole we have the wave movement of Fig.9.

      A          -------------------------->
      |x
      A-,                               ,---,
      |  `\                           /'     `\
      |    `                         '         `
 ~E{*}|     '            _ - '/^,   '           `            _ - '^\
      |      \       _-'`   /'  |  /             \       _-'`       |
      |       \  _,-'     /~B{*}` /               \  _,-'            z
      *-------_+'--------'-------+--------------- _+'---------------->
    /~B{*}_-'`  \       |       / ,           _-'`  \               /
  /'  _,-~       \      |~E{*} /  |       _,-'       \             /
l/,-'`            ,     |     ,    \__,-'`            ,           ,
y                  ,    |    ,                         ,         ,
                    \.  |  ,/                           \.     ,/
                      `-v-'                               `---'

             Fig.9. Propagation of an electromagnetic wave.

The physical interpretation presented above is raw but properly shows that 
why in each point of the medium of the wave propagation ~E{*} and ~B{*}
are perpendicular to each other in such a way that ~E{*}<cross>~B{*} shows
the direction of propagation of the wave. From the above discussions we can 
deduce that the maximums and minimums of ~E{*} and ~B{*} occur 
simultaneously, because otherwise at some part of a period the direction 
of propagation of the wave would be reversed and this is not possible
(or at least this is not the case in the discussions presented in this 
article, on supposition).

VII. Sinusoidal waves and Fresnel coefficients
----------------------------------------------
We can rewrite the wave equations (31) and (32) as

<round>{2}~E{*}/<round>t{2} = v{2}<del>{2}~E{*}   )
                                                  >             (40)
<round>{2}~B{*}/<round>t{2} = v{2}<del>{2}~B{*}   )

in which v=(<mu>{2}<epsilon><epsilon>'aa'){-1/2} is the speed of propagation
of the wave. (In this article we suppose that the mediums of propagation of
the electromagnetic wave are isotropic such that the speed of the wave in 
all the directions is the same.) A particular solution to these wave 
equations for, for example, the x component of the vector ~E{*} is 
E[x]{*}=f[1](^u.~r-vt)+f[2](^u.~r+vt) which exhibits a plane wave moving 
in the sense of the unit vector ^u with the absolute speed |v|. Then
E[x]{*}(~r,t)=E[x0]{*}cosk(^u.~r-vt) can be a solution, and since 
cosk(^u.~r-vt)=cosk(^u.~r+(2<pi>/k)-vt), we conclude that <lambda>=2<pi>/k
represents the spatial period of the wave or the wave length, and
k=2<pi>/<lambda>, which is the number of the wave lengths in distance 2<pi>,
is called as the wave number, and with the definition of the angular
frequency of the wave, <omega>, and the period of the wave, T, and the
frequency of the wave, <nu>, by the relation
<omega>=kv=2<pi>v/<lambda>=2<pi>/(<lambda>/v)=2<pi>/T=2<pi><nu>
and with the definition of k^u=~k (as the wave vector) we shall have
E[x]{*}=E[x0]{*}cos(~k.~r-<omega>t) (which is also equal to the real part 
of the E[x0]{*}exp(-i(<omega>t-~k.~r))). Then we suppose that we have
~B{*}=B[x0]{*}cos(~k.~r-<omega>t)^i+B[y0]{*}cos(~k.~r-<omega>t)^j+
B[z0]{*}cos(~k.~r-<omega>t)^k and E{*}=E[x0]{*}cos(~k.~r-<omega>t)^i+
E[y0]{*}cos(~k.~r-<omega>t)^j+E[z0]{*}cos(~k.~r-<omega>t)^k. 
From the equations (30), by relevant differentiations, it is easily 
obtained that                                      

~k.~E{*} = 0 , ~k.~B{*} = 0 ,                    )
~k<cross>~E{*} = <mu><epsilon>'a'<omega>~B{*},   >              (41)
~k<cross>~B{*} = -<mu><epsilon>a<omega>~E{*}     )

which indicates that the orthogonal system (~E{*}, ~B{*}, ~k) is right-handed.
We consider a medium (generally the vacuum (we should notice that the vacuum
is not empty of the mentioned tiny dipoles, while the free space is so)) as 
the reference medium and we show the speed of the electromagnetic wave in it 
by c. We show the ratio of the c to the speed of the electromagnetic wave 
in each medium by n and we call it as the refractive index of that medium
(n=c/v and then v=c/n). Therefore, we have k=<omega>/v=n<omega>/c. 
Considering this matter, from the last two relations of (41) we shall have:

~B{*} = n/(<mu><epsilon>'a'c)^u<cross>~E{*}      )
                                                 >              (42)
~E{*} = -n/(<mu><epsilon>ac)^u<cross>~B{*}       )

It is obvious that n/(<mu><epsilon>'a'c).n/(<mu><epsilon>ac)=1.
It is proper here to see the following deduction:

(42) ==> B{*}/E{*} = 1/(<mu><epsilon>'a'v)
= (<mu>{2}<epsilon><epsilon>'aa'){1/2}/(<mu><epsilon>'a')
= (<mu><epsilon>a){1/2}/(<mu><epsilon>'a'){1/2} ==>
(<mu><epsilon>'a'){1/2}B{*}-(<mu><epsilon>a){1/2}E{*} = 0 ==>
<mu><epsilon>'a'B{*}{2}+<mu><epsilon>aE{*}{2} 
= 2(<mu>{2}<epsilon><epsilon>'aa'){1/2}E{*}B{*} ==>
~E{*}<cross>~B{*} = (1/2<mu><epsilon>'a'~B{*}.~B{*} +
1/2<mu><epsilon>a~E{*}.~E{*})^u/(<mu>{2}<epsilon><epsilon>'aa'){1/2} ==>
1/<mu>~E{*}<cross>~B{*}=(1/2<epsilon>'~B{*}.~B+1/2<epsilon>~E{*}.~E)v^u
==> ~S = u~v ,                                                  (43)

and we know that the relation (43) is an expectable relation, comparable 
with the relation ~J = <rho>[E]~v.

Now we proceed to the boundary conditions. We have <del>.~B{*}=0. Considering
a pillbox-shaped surface, as in Fig.10, which its height is infinitesimal,
and applying the divergence theorem over the volume enclosed by this surface
we obtain

<integral over V><del>.~B{*}dv
=<integral over S[1]>~B{*}.^n[1]da+<integral over S[2]>~B{*}.^n[2]da )
                                                                     >
                                                     <del>.~B{*} = 0 )

==> <integral over S[1]>~B{*}.^n[1]da 
= <integral over S[2]>~B{*}.(-^n[2])da

and since S[1]=S[2] and ^n[1]=-^n[2], considering ^n[1] as ^n we shall
have B[1n]{*}=B[2n]{*}.
                                  A
                                  |^n[1]
                               _,-|--,_
                              (   |S[1])
                         _____|` ---- '|________1___
                       /'     |` ---- '|        2 /'
                     /'    S[2]` ---- '         /'
                   /'             |^n[2]      /'
                 /'               V         /'
               /__________________________/'

      Fig.10. A pillbox-shaped surface at the interface between
              two media, used to obtain continuity of the normal
              components of the fields across the interface.

In a similar manner we shall obtain E[1n]{*}=E[2n]{*}. Then the normal 
components of the fields are continuous across the boundary interface. Now 
from the equation <del><cross>~B{*} = <mu><epsilon>a<round>~B{*}/<round>t 
we have <integral over S><del><cross>~B{*}.^nda =
<integral over S><mu><epsilon>a<round>~E{*}/<round>t.^nda in which S is the
area enclosed by the loop in Fig.11.

                         ,--------->---------,
                         |_,----'''^'''----,_|
                   _,--''`---------<---------'`'--,_1
              _,-'`                                 2'-,_

       Fig.11. A rectangular path at the interface between two media,
               used to obtain continuity of the tangential components
               of the fields across the interface.

If the width of this rectangular loop is infinitesimal, by applying Stokes'
theorem and with attention to this fact that the flat area enclosed by the
loop approaches zero and therefore the second integral approaches zero, we
shall obtain B[1t]{*}=B[2t]{*}. In a similar manner we shall obtain
E[1t]{*}=E[2t]{*}. (It is necessary to notice that the orientation of this
loop is optional.) Then the tangential components of the fields are 
continuous across the boundary interface too, and then altogether we should
say that the dynamic fields connected to an electromagnetic wave are
continuous across the boundary interfaces.

With attention to Fig.12 we suppose that the electrodynamic fields of the
incident, reflected, and transmitted waves of a plane electromagnetic wave
at the boundary of a dielectric with the refractive index n[2], set in a
space with the refractive index n[1], are as follows:

~E[1]{*} =                                                  )
~E[10p]{*}cos(~k[1].~r-<omega>[1]t) +                       |
~E[10s]{*}cos(~k[1].~r-<omega>[1]t)                         |
~E[1']{*} =                                                 |
~E[1'0p]{*}cos(~k[1'].~r-<omega>[1']t) +                    >   (44)
~E[1'0s]{*}cos(~k[1'].~r-<omega>[1']t                       |
~E[2]{*} =                                                  |
~E[20p]{*}cos(~k[2].~r-<omega>[2]t) +                       |
~E[20s]{*}cos(~k[2].~r-<omega>[2]t)                         )

                                A
           .__                  | x         .__
           |\                   |           |\
      ~k[1'] `\ ~B[1']{*}       |             `\
               ,0               |       ~E[2]{*}`\   ~k[2]__.
           ,-'`  `\        n[1] | n[2]            `\    _,/'|
       _,-`        `\           |                   `0-'
    !_' ~E[1']{*}    `\         |                ,-' ~B[2]{*}
                       `\       |            _,-'
                         `\     |         ,-'
                           `\   |     _,-'
                  <theta>[1']`\ | _,-'<theta>[2]
             ------------------`|---------------------------------->
                    <theta>[1]/'|                                 z
                            /'  |
                          /'    |
    .__              __./'      |
    |',               /|        |
       `'-_    ~k[1]/'          |
   ~E[1]{*}`-,_   /'            |
               `0'
               ~B[1]{*}

    Fig.12. Incident, reflected, and transmitted (or refracted) waves
            at the boundary. Vector ~B{*} is directed out of the paper.

These three fields must be in phase in ~r=~0 in different times, then 
<omega>[1]=<omega>[1']=<omega>[2]=<omega>; and they must be also in phase
in each point on the boundary interface, z=0, at t=0, so we have the    
relations ~k[1].~r=~k[1'].~r=~k[2].~r on the boundary, from which the three
consequences of the coplanarity of the wave vectors, the law of reflection,
and Snell's law are easily obtained in the manner followed by many of the
electromagnetic and optics books. Now we want to obtain the Fresnel
coefficients. Substituting the first relation of (42) into the boundary 
condition ~B[1]{*}+~B[1']{*}=~B[2]{*} we obtain 
n[1]/(<mu>[1]<epsilon>[1]'a[1]')(^u[1]<cross>~E[1]{*}+^u[1']<cross>~E[1']{*})
= n[2]/(<mu>[2]<epsilon>[2]'a[2]')^u[2]<cross>~E[2]{*} which with the
cross product of the unit vector ^n=^k, which is normal to the interface,
multiplied by the two sides of the recent relation we shall have n[1]/(<mu>[
1]<epsilon>[1]'a[1]')^n<cross>(^u[1]<cross>~E[1]{*}+^u[1']<cross>~E[1']{*})=
n[2]/(<mu>[2]<epsilon>[2]'a[2]')^n<cross>(^u[2]<cross>~E[2]{*}). Expanding
the vector products in the recent relation, and considering only the 
components normal to the plane of incidence (ie the plane of ^k[1] and ^n),
which are distinguished by the subscript s, and with attention to this fact
that <theta>[1]=<theta>[1'], we conclude that

n[1]/(<mu>[1]<epsilon>[1]'a[1]')cos<theta>[1](~E[1s]{*}-~E[1's]{*})
= n[2]/(<mu>[2]<epsilon>[2]'a[2]')cos<theta>[2]~E[2s]{*}.       (45)

The other boundary condition, ie ~E[1]{*}+~E[1']{*}=~E[2]{*}, also 
necessitates having separately the relation: 

~E[1s]{*}+E[1's]{*}=~E[2s]{*}                                   (46)

By solvingh the equations (45) and (46) for ~E[1's]{*} and ~E[2s]{*},
we shall have

~E[1's]{*}=r[12s]~E[1s]{*}  &  ~E[2s]{*}=t[12s]~E[1s]{*}        (47)

in which

r[12s] = 
(n[1]/(<mu>[1]<epsilon>[1]'a[1]')cos<theta>[1]-n[2]/(<mu>[2]<epsilon>[2]'a[
2]')cos<theta>[2])/(n[1]/(<mu>[1]<epsilon>[1]'a[1]')cos<theta>[1]+n[2]/(<
mu>[2]<epsilon>[2]'a[2]')cos<theta>[2])                         (48)

and

t[12s] = 
(2n[1]/(<mu>[1]<epsilon>[1]'a[1]')cos<theta>[1])/(n[1]/(<mu>[1]<
epsilon>[1]'a[1]')cos<theta>[1]+n[2]/(<mu>[2]<epsilon>[2]'a[2]'
)cos<theta>[2]).                                                (49)

And also by substituting the second relation of (42) into the boundary 
condition ~E[1]{*}+~E[1']{*}=~E[2]{*} and performing similar operations and
considering the other boundary condition, ie ~B[1]{*}+~B[1']{*}=~B[2]{*}, we
shall obtain

~B[1's]{*}=r[12p]~B[1s]{*}  &  ~B[2s]{*}=t[12p]~B[1s]{*}        (50)

in which

r[12p] =
(n[1]/(<mu>[1]<epsilon>[1]a[1])cos<theta>[1]-n[2]/(<mu>[2]<epsilon
>[2]a[2])cos<theta>[2])/(n[1]/(<mu>[1]<epsilon>[1]a[1])cos<theta>[
1]+n[2]/(<mu>[2]<epsilon>[2]a[2])cos<theta>[2])                 (51)

and

t[12p] =
(2n[1]/(<mu>[1]<epsilon>[1]a[1])cos<theta>[1])/(n[1]/(<mu>[1]<epsilon
>[1]a[1])cos<theta>[1]+n[2]/(<mu>[2]<epsilon>[2]a[2])cos<
theta>[2]).                                                     (52)

Also, for the components parallel to the plane of incidence, which are
distinguished by the subscript p, we have

~E[2p]{*} = -n[2]/(<mu>[2]<epsilon>[2]a[2]c)^u[2]<cross>~B[2s]{*} =
-n[2]/(<mu>[2]<epsilon>[2]a[2]c)^u[2]<cross>t[12p]~B[1s]{*} =
n[2]n[1]t[12p]/(<mu>[2]<epsilon>[2]a[2]c<mu>[1]<epsilon>[1]'a[1]'c)^u[2]<
cross>(^u[1]<cross>~E[1p]{*}) ==> E[2p]{*} = n[2]n[1]/(<mu>[2]<epsi
lon>[2]a[2]c<mu>[1]<epsilon>[1]'a[1]'c)|t[12p]|E[1p]{*},        (53)
~B[2p]{*} = n[2]/(<mu>[2]<epsilon>[2]'a[2]'c)^u[2]<cross>~E[2s]{*} =
n[2]/(<mu>[2]<epsilon>[2]'a[2]'c)^u[2]<cross>t[12s]~E[1s]{*} =
-n[2]n[1]t[12s]/(<mu>[2]<epsilon>[2]'a[2]'c<mu>[1]<epsilon>[1]a[1]c)^u[2]<
cross>(^u[1]<cross>~B[1p]{*}) ==> B[2p]{*} = n[2]n[1]/(<mu>[2]<epsi
lon>[2]'a[2]'c<mu>[1]<epsilon>[1]a[1]c)|t[12s]|B[1p]{*},        (54)
~E[1'p]{*} = -n[1]/(<mu>[1]<epsilon>[1]a[1]c)^u[1']<cross>~B[1's]{*} =
-n[1]/(<mu>[1]<epsilon>[1]a[1]c)^u[1']<cross>r[12p]~B[1s]{*} =
-r[12p]^u[1']<cross>(^u[1]<cross>~E[1p]{*}) ==>
E[1'p]{*} = |r[12p]|E[1p]{*},                                   (55)
~B[1'p]{*} = n[1]/(<mu>[1]<epsilon>[1]'a[1]'c)^u[1']<cross>~E[1's]{*} =
n[1]/(<mu>[1]<epsilon>[1]'a[1]'c)^u[1']<cross>r[12s]~E[1s]{*} =
-r[12s]^u[1']<cross>(^u[1]<cross>~B[1p]{*}) ==>
B[1'p]{*} = |r[12s]|B[1p]{*}.                                   (56)

For obtaining more confidence to the correctness of the above relations
and expressions obtained for Fresnel coefficients, it is proper to examine
them when <theta>[1]=<theta>[2]=0, because the isotropy of the mediums, 
which is our supposition in this article, necessitates having
(E[1'p]{*}/E[1p]{*})=(E[1's]{*}/E[1s]{*} or |r[12p]|=|r[12s]| in this state,
ie we should have an identity in the form of r[12p]=r[12s] or r[12p]=-r[12s]
when <theta>[1]=<theta>[2]=0. A simple investigation shows that only the
relation r[12p]=-r[12s] is an identity when <theta>[1]=<theta>[2]=0 (in this
investigation we need the obvious relation n[1]v[1]=n[2]v[2]). 
Likewise, the isotropy of the mediums necessitates having
(E[2p]{*}/E[1p]{*})=(E[2s]{*}/E[1s]{*}) or 
(n[2]n[1]/(<mu>[2]<epsilon>[2]a[2]c<mu>[1]<epsilon>[1]'a[1]'c))|t[12p]|=
|t[12s]| in the case of <theta>[1]=<theta>[2]=0, which again a simple 
investigation shows that we have the identity 
t[12s]=(n[2]n[1]/(<mu>[2]<epsilon>[2]a[2]c<mu>[1]<epsilon>[1]'a[1]'c))t[12p]
when <theta>[1]=<theta>[2]=0. The isotropy of the mediums in the case of 
<theta>[1]=<theta>[2]=0 for ~B{*} also necessitates the identities
|r[12s]|=|r[12p]| and 
(n[2]n[1]/(<mu>[2]<epsilon>[2]'a[2]'c<mu>[1]<epsilon>[1]a[1]c))|t[12s]|=
|t[12p]|, but as said before we have the identity r[12s]=-r[12p] for the
first and a simple investigation shows that we have the identity
t[12p]=(n[2]n[1]/(<mu>[2]<epsilon>[2]'a[2]'c<mu>[1]<epsilon>[1]a[1]c))t[12s]
for the second, both when <theta>[1]=<theta>[2]=0.

With attention to the relation n/(<mu><epsilon>a)=c{2}<mu><epsilon>'a'/n 
we can write

r[12p]=((<mu>[1]<epsilon>[1]'a[1]'/n[1])cos<theta>[1]-(<mu>[2]<epsilon
>[2]'[a[2]'/n[2])cos<theta>[2])/((<mu>[1]<epsilon>[1]'a[1]'/n[1])cos<
theta>[1]+(<mu>[2]<epsilon>[2]'a[2]'/n[2])cos<theta>[2])        (57)

and

t[12p]=(2(<mu>[1]<epsilon>[1]'a[1]'/n[1])cos<theta>[1])/((<mu>[1]<ep
silon>[1]'a[1]'/n[1])cos<theta>[1]+(<mu>[2]<epsilon>[2]'a[2]'/n[
2])cos<theta>[2]).                                              (58)

Now with definition of n[E]=n/(<mu><epsilon>'a') we have
                                                         
         n[E1]cos<theta>[1]-n[E2]cos<theta>[2]           )
r[12s] = -------------------------------------     ,     |
         n[E1]cos<theta>[1]+n[E2]cos<theta>[2]           |
                                                         >      (59)
                 2n[E1]cos<theta>[1]                     |
t[12s] = -------------------------------------           |
         n[E1]cos<theta>[1]+n[E2]cos<theta>[2]           )
        
and

         n[E2]cos<theta>[1]-n[E1]cos<theta>[2]           )
r[12p] = -------------------------------------     ,     |
         n[E2]cos<theta>[1]+n[E1]cos<theta>[2]           |
                                                         >      (60)
                 2n[E2]cos<theta>[1]                     |
t[12p] = -------------------------------------     .     |
         n[E2]cos<theta>[1]+n[E1]cos<theta>[2]           )

And with attention to the relation n/(<mu><epsilon>'a')=
c{2}<mu><epsilon>a/n and definition of n[B]=n/(<mu><epsilon>a) we have

         n[B2]cos<theta>[1]-n[B1]cos<theta>[2]           )
r[12s] = -------------------------------------     ,     |
         n[B2]cos<theta>[1]+n[B1]cos<theta>[2]           |
                                                         >      (61)
                 2n[B2]cos<theta>[1]                     |
t[12s] = -------------------------------------           |
         n[B2]cos<theta>[1]+n[B1]cos<theta>[2]           )

and

         n[B1]cos<theta>[1]-n[B2]cos<theta>[2]           )
r[12p] = -------------------------------------     ,     |
         n[B1]cos<theta>[1]+n[B2]cos<theta>[2]           |
                                                         >      (62)
                 2n[B1]cos<theta>[1]                     |
t[12p] = -------------------------------------     .     |
         n[B1]cos<theta>[1]+n[B2]cos<theta>[2]           )

and also 
n[2]n[1]/(<mu>[2]<epsilon>[2]'a[2]'c<mu>[1]<epsilon>[1]a[1]c) =
n[B1]n[E2]/c{2} and
n[2]n[1]/(<mu>[2]<epsilon>[2]a[2]c<mu>[1]<epsilon>[1]'a[1]'c) =
n[E1]n[B2]/c{2}.

Before continuing I should mention a point. Attention to this fact that in
the relations (5) and (7) of the fundamental relations k" which is 
proportional to <mu> shows the relation between two electric and magnetic
charges only when they are moving relative to each other (and the situation
is not like the static situation in which two electric charges or two
magnetic charges, which are stationary relative to each other, can polarize
the space between themselves and depending on this polarization of the
medium can change k and k' in fact) states that <mu> should be the same for 
all the mediums and in fact it should be a world constant. Now considering 
this matter and in fact accepting it and with attention to the relation
n[1]sin<theta>[1]=n[2]sin<theta>[2], if the magnetic characteristics of the
two mediums are the same, ie if we have a[1]'=a[2]' and 
<epsilon>[1]'=<epsilon>[2]', then we shall see that

       n[1]cos<theta>[1]-n[2]cos<theta>[2]  sin(<theta>[2]-<theta>[1])
r[12s]= --------------------------------- = -------------------------,
       n[1]cos<theta>[1]+n[2]cos<theta>[2]  sin(<theta>[2]+<theta>[1])
                                                                  (63)

                2n[1]cos<theta>[1]         2cos<theta>[1]sin<theta>[2]
t[12s]= --------------------------------- = -------------------------,
       n[1]cos<theta>[1]+n[2]cos<theta>[2]  sin(<theta>[1]+<theta>[2])
                                                                  (64)

       n[2]cos<theta>[1]-n[1]cos<theta>[2]  tan(<theta>[1]-<theta>[2])
r[12p]= --------------------------------- = -------------------------,
       n[2]cos<theta>[1]+n[1]cos<theta>[2]  tan(<theta>[1]+<theta>[2])
                                                                  (65)
                2n[2]cos<theta>[1]
t[12p]= ----------------------------------
       n[2]cos<theta>[1]+n[1]cos<theta>[2]

                         sin(2<theta>[1])
      = ----------------------------------------------------,     (66)
        sin(<theta>[1]+<theta>[2])cos(<theta>[1]-<theta>[2])

and since 1/(<epsilon>a)=c{2}<mu>{2}a'<epsilon>'/n{2}, <mu>[1]=<mu>[2],
<epsilon>[1]'=<epsilon>[2]' and a[1]'=a[2]', we conclude that
n[2]n[1]/(<mu>[2]<epsilon>[2]a[2]c<mu>[1]<epsilon>[1]'a[1]'c)=n[1]/n[2] and
n[2]n[1]/(<mu>[2]<epsilon>[2]'a[2]'c<mu>[1]<epsilon>[1]a[1]c)=n[2]/n[1] and
(n[1]/n[2])t[12p]=2cos<theta>[1]sin<theta>[2]/(sin(<theta>[1]+
<theta>[2])cos(<theta>[1]-<theta>[2])) and 
(n[2]/n[1])t[12s]=sin(2<theta>[1])/sin(<theta>[1]+<theta>[2]) too. In a
similar manner we have the following relations for the condition in which
the electric characteristics of the two mediums are the same, ie when
a[1]=a[2] and <epsilon>[1]=<epsilon>[2]:

       n[2]cos<theta>[1]-n[1]cos<theta>[2]  tan(<theta>[1]-<theta>[2])
r[12s]= --------------------------------- = -------------------------,
       n[2]cos<theta>[1]+n[1]cos<theta>[2]  tan(<theta>[1]+<theta>[2])
                                                                  (67)

                2n[2]cos<theta>[1]
t[12s]= ----------------------------------
       n[2]cos<theta>[1]+n[1]cos<theta>[2]

                         sin(2<theta>[1])
      = ---------------------------------------------------,      (68)
        sin(<theta>[1]+<theta>[2])cos(<theta>[1]-<theta>[2])

       n[1]cos<theta>[1]-n[2]cos<theta>[2]  sin(<theta>[2]-<theta>[1])
r[12p]= --------------------------------- = -------------------------,
       n[1]cos<theta>[1]+n[2]cos<theta>[2]  sin(<theta>[2]+<theta>[1])
                                                                  (69)

                2n[1]cos<theta>[2]              sin(2<theta>[2])
t[12p]= --------------------------------- = -------------------------,
       n[1]cos<theta>[1]+n[2]cos<theta>[2]  sin(<theta>[1]+<theta>[2])
                                                                  (70)

and also 
n[2]n[1]/(<mu>[2]<epsilon>[2]a[2]c<mu>[1]<epsilon>[1]'a[1]'c)=n[2]/n[1] and
n[2]n[1]/(<mu>[2]<epsilon>[2]'a[2]'c<mu>[1]<epsilon>[1]a[1]c)=n[1]/n[2] and 
then (n[2]/n[1])t[12p]=2sin<theta>[1]cos<theta>[2]/sin(<theta>[1]+<theta>[2])
and (n[1]/n[2])t[12s]=2cos<theta>[1]sin<theta>[2]/(sin(<theta>[1]+
<theta>[2])cos(<theta>[1]-<theta>[2])).

In Brewster's incident angle <theta>[1]=<theta>[B] in which 
<theta>[B]+<theta>[2]=<pi>/2 and then 1/tan(<theta>[B]+<theta>[2])=0 if the
magnetic characteristics of the two mediums are the same, we shall have
r[12p]=0 and so in the reflection we can only have an electromagnetic wave 
which the vibration of its ~E{*} is perpendicular to the plane of incidence;
and if the electric characteristics of the two mediums are the same, we shall
have r[12s]=0 and so in the reflection we can only have an electromagnetic 
wave which the vibration of its ~E{*} lies in the plane of incidence. If the
laboratory works show that in Brewster's angle we have the first state in
the mediums under experimentation, we should conclude that in these mediums
the magnetic characteristics are the same and are not changed from a medium
to another medium practically.

Finally we pay our attention to the energy or the intensity of the
electromagnetic wave. We have

~S=(~E{*}<cross>~B{*})/<mu>=(~E{*}<cross>(n/(<mu><epsilon>'a'c)^u<
cross>~E{*}))/<mu>=n/(<mu>{2}<epsilon>'a'c)E{*}{2}^u ==>
(~S)[av]=1/2 n/(<mu>{2}<epsilon>'a'c)E[0]{*}{2}^u,
~S=(~E{*}<cross>~B{*})/<mu>=((-n/(<mu><epsilon>ac)^u<cross>~B{*})<
cross>~B{*})/<mu>=n/(<mu>{2}<epsilon>ac)B{*}{2}^u ==>
(~S)[av]=1/2 n/(<mu>{2}<epsilon>ac)B[0]{*}{2}^u.

On definition we have the following relations:

       (-^n).(~S[1's])[av]                ^n.(~S[2s])[av]
R[s] = -------------------    ,    T[s] = ---------------    ,  (71)
         ^n.(~S[1s])[av]                  ^n.(~S[1s])[av]

       (-^n).(~S[1'p])[av]                ^n.(~S[2p])[av]
R[p] = -------------------    ,    T[p] = ---------------    ,  (72)
         ^n.(~S[1p])[av]                  ^n.(~S[1p])[av]

and in terms of the Fresnel coefficients we shall have


R[s]= r[12s]{2} ,                                               (73)
                                                                       
     1/2(n[2]/(<mu>[2]{2}<epsilon>[2]'a[2]'c))cos<theta>[2]           
T[s]=------------------------------------------------------t[12s]{2},
     1/2(n[1]/(<mu>[1]{2}<epsilon>[1]'a[1]'c))cos<theta>[1]           
                                                                (74)
R[p]= r[12p]{2} ,                                               (75)

     1/2(n[2]/(<mu>[2]{2}<epsilon>[2]'a[2]'c))cos<theta>[2]        n[
T[p]=------------------------------------------------------ . -------
     1/2(n[1]/(<mu>[1]{2}<epsilon>[1]'a[1]'c))cos<theta>[1]   <mu>[2]

2]{2}n[1]{2}
---------------------------------------------------------
{2}<epsilon>[2]{2}a[2]{2}c{2}<mu>[1]{2}<epsilon>[1]'{2}a[


----------t[12p]{2}.                                            (76)
1]'{2}c{2}


By using some simple algebraic operations and accepting that <mu> is a 
world onstant it is easily seen that we have always

R[s]+T[s] = 1  ,  R[p]+T[p] = 1  .                              (77)

VIII. Conclusion
----------------
As it was seen, by using an obvious and simple supposition, mentioned in
the introduction and formed mathematically in the fundamental relations,
all of the relations of the electromagnetic theory were modified and 
perfected in a sense consistent with the classical physics, and in 
addition to this, new relations and theories were presented for showing
themselves in practice and experiment.

What is perhaps more important than the carefulness in the details of the
presented discussions is attention to the validity and stability of the
front opened for defending the classical physics and mathematical logic
of the classical mechanics.

Hamid V. Ansari