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Started by esfahan
Wed, 27 Jan 1999 00:00
Mistakes in Electrostatics; Dreadful consequences in Modern Physics
Author: esfahan
Date: Wed, 27 Jan 1999 00:00
Date: Wed, 27 Jan 1999 00:00
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With respect for the society of the world science:
The purpose of publication of the following article is its dissemination
and taking the ideas presented in it under discussion. I request to
record it and put it at others' disposal too.
I am greatly interested in and insistent on that these ideas to be disscused
and criticized profoundly. Meanwhile, I am quite aware of the so much
importance of these ideas and the alterations which they will cause in
physics, and then contrary to all the difficulties I won't desist from
their publication (even repeatedly) unless their incorrectness is proved
scientifically. My aim is not acquisition of fame but finding out the fact.
This article is from the set of articles collected in a book entitled
"Great mistakes of the physicists. Isn't physics going astray?". These
articles self-compatibly present a logical and unique feature of physics
without any nonclassical presupposition, and as I pointed I am actively
making them better as far as my possibilities permit.
You can have the last edition of this book in your e-mail box free of
charge if you send me your request and e-mail address. (You can also
purchase the book itself by paying its cost.) Abstracts of the articles
of this book is on the following Web address:
http://magna.com.au/~prfbrown/news97_k.html
Please study the biography of the author at the end of this article
after studying the article.
Since I have difficulty in regular accessing to Internet I'll be thankful
if you send your replies to <jc626@cleveland.Freenet.Edu> too.
Hamid Vasigh Ansari
NOTE:
~~~~
I recommend you not to forget to study the last section (IV) of this
article even as an independent part separate from the other sections.
There you can see interesting material about the experiments for
determination of charge and mass of the electron.
We use the following 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.
We show integral around a closed space as <circulation>.
In a capital Greek letter, the word "cap." is written.
(From the 6th article of the book)
""""""""""""""""""""""""""""""""""
Mistakes in Electrostatics; Dreadful consequences in Modern Physics
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Abstract
--------
It is shown that there exist a uniqueness theorem, stating that the
charges given to a constant configuration of conductors take a unique
distribution, which contrary to what is believed does not have any relation
to the uniqueness theorem of electrostatic potential. Using this thorem
we obtain coefficients of potential analytically. We show that a simple
carelessness has caused the famous formula for the electrostatic potential
to be written as U=1/2<integral>~D.~Edv while its correct form is
U=1/2<integral>~D.~E[<rho>]dv in which ~E[<rho>] is the electrostatic
field arising only from the external charges not also from the
polarization charges.
Considering the above-mentioned material it is shown that, contrary to
the current belief, capacitance of a capacitor does not at all depend on
the dielectric used in it and depends only on the configuration of its
conductors. We proceed to correct some current mistakes resulted from
the above-mentioned mistakes, eg electrostatic potential energy of and the
inward force exerted on a dielectric block entering into a parallel-plate
capacitor are obtained and compared with the wrong current ones.
It is shown that existence of dielectric in the capacitor of a circuit
causes attraction of more charges onto the capacitor because of the
polarization of the dielectric. Then, in electric circuits we should
consider the capacitor's dielectric as a source of potential not think
wrongly that existence of dielectric changes the capacitor's capacitance.
Difference between these two understandings are verified completely during
some examples, and some experiments are proposed for testing the theory.
For example it is shown that contrary to what the current theory predicts,
resonance frequency of a circuit of RLC will increase by inserting
dielectric into the capacitor.
It is also shown that contrary to this current belief that the
electrostatic potential difference between the two conductors of a
capacitor is the same potential difference between the two poles of the
battery which has charged it, the first is twofold compared with the
second. We see the influence of this in the experiments performed for
determination of charge and mass of the electron.
I. Introduction
---------------
In the current electrostatic discussions it is stated that a solution
of Laplace's equation which fits a set of boundary conditions is unique,
and while this matter has not been proved in the case that these boundary
conditions are the charges on the boundaries, the known charges on the
boundaries are taken as boundary conditions. First section of this
article solves this problem after which obtains the coefficient of
potential, while in the current electromagnetic books these coefficients
are obtained by using the above mentioned unproved generalization of
the boundary conditions which incorrectness of this way is also shown.
The relation U=1/2<integral over V>~D.~Edv for the electrostatic potential
energy of a system is a quite familiar equation to every physicist,
but a careful scrutiny shows an existent undoubted mistake in this
equation. This mistake is easily arising from this fact that in the
process of obtaining this equation, while accepting that <del>.~D=<rho>
where <rho> is the external electric charge density, it is forgotten that
in the primary equation of the electrostatic potential energy of the
system the potential arising only from this <rho>, <phi>[<rho>], not also
from the polarization charges be taken into account resulting in
considering ~E (obtained from -<del><phi>) instead of ~E[<rho>] (obtained
from -<del><phi>[<rho>]) which is the electrostatic field arising only
from <rho> not also from the polarization. This careful scrutiny is
presented in the third section of this article. A great part of this
section proceeds to some consequences of this same mistake including
this current belief that the capacitance of a capacitor depends on its
dielectric, while we shall prove that this is not at all the case and
it depends only on the form of the configuration of the conductors of
the capacitor.
To another much simple and obvious current mistake is paid in the last
section: We connect a battery, which the potential difference between
its poles is <cap. delta><phi>, to the two plates of an uncharged
capacitor until it will be charged. Then, what is the electrostatic
potential difference between the plates of the charged capacitor? All
the current literature on the subject answer that this electrostatic
potential difference is the same potential difference between the poles
of the battery, <cap. delta><phi>, while this is not the case and is
equal to 2<cap. delta><phi>.
As it is seen, the above current mistakes some of which being fundamental
are totally in bases of the subject of Electromagnetism, and cannot
be ignored, because not only are much widespread and taught in all the
universities but also some of them are basis for some subsequent
deductions in other branches of physics. This matter shows that in the
progress of physics the attention should not be only to its rapidity
but also to its profundity, otherwise, as in the case of this article,
sometimes some of the obvious mistakes remain hidden from the physicists'
view yielding probably very other mistake consequences.
II. Another uniqueness theorem in Electrostatics
------------------------------------------------
II.A. Uniqueness theorem of charge distribution in conductors
-------------------------------------------------------------
In solving electrostatic problems there is a uniqueness theorem that
distinctly states that when the electrostatic potential or the normal
component of its gradient is given in each point of the bounding surfaces
then if the potential is given in at least one point, the solution of
Laplace's equation is uique, and otherwise we may add any constant to a
solution of this equation. Unfortunately, sometimes negligence is seen
in careful applying of the quite clear stated above boundary conditions.
For instance without any reason the charges of bounding surfaces are
taken as boundary conditions in terms of which the above theorem is
applied in obtaining coefficients of potential of a system of conductors.
The reasoning being used is this (see Foundations of Electromagnetic
Theory by Reitz, Milford and Christy, Addison-Wesley, 1979):
"Suppose there are N conductors in fixed geometry. Let all the conductors
be uncharged except conductor j, which bears the charge Q[j0]. The
appropriate solution to Laplace's equation in the space exterior to
the conductors will be given the symbol <phi>{(j)}(x,y,z) and the potential
of each of the conductors will be indicated by <phi>{(j)}[1], <phi>{(j)}[2],
..., <phi>{(j)}[j], ....,<phi>{(j)}[N]. Now let us change the charge
of the jth conductor to <lambda>Q[j0]. The function <lambda><phi>{(j)}(x,y,z)
satisfies Laplace's equation, since <lambda> is a constant; that the
new boundary conditions are satisfied by this function may be seen from
the following argument. The potential at all points in space is multiplied
by <lambda>; thus all derivatives (and in particular the gradient) of the
potential are multiplied by <lambda>. Because <sigma>=<epsilon>[0]E[n],
it follows that all charge densities are multiplied by <lambda>. Thus the
charge of the jth conductor is <lambda>Q[j0] and all other conductors
remain uncharged. A solution of Laplace's equation which fits a particular
set of boundary conditions is unique; therefore we have found the correct
solution, <lambda><phi>{(j)}(x,y,z) to our modified problem. The conclusion
we draw from this discussion is that the potential of each conductor is
proportional to the charge Q[j] of conductor j, that is
<phi>{(j)}[i]=p[ij]Q[j], (i=1,2,...,N) where p[ij] is a constant which
depends only on the geometry."
The fault may be found in this reasoning is arising from the same incorrect
distinction of boundary conditions. This fault is that a solution to
Laplace's equation other than <lambda><phi>{(j)} can be found such that it
can make the charge of the jth conductor <lambda>fold retaining all other
conductors uncharged. This solution can be <lambda><phi>{(j)}(x,y,z)+c
for a non-zero constant c. It is obvious that its gradient and therefore
<sigma>=<epsilon>[0]E[n] arising from it compared with before are
<lambda>fold and then the charge of the jth conductor will be
<lambda>fold while all other conductors remain uncharged. But this
solution is no longer proportional to the charge of the jth conductor,
Q[j], ie we won't have <phi>{(j)}(x,y,z)=p[ij]Q[j].
In order to clear obviously that the uniqueness theorem of potential
does not include boundary conditions on charges, suppose that there is an
initially uncharged conductor. We then give it some charge. We want to see
when the given charge is definite whether potential function outside the
conductor will or won't be determined uniquely by this theorem. We say
that the given charge distributes itself onto the surface of the conductor
and remain fixed causing that the potential of the equipotential surface
of the conductor to become specified. With specifying of the conductor
potential, potential function outside the conductor is determined uniquely
according to the theorem. But important for us is knowing that whether
form of the charge distribution onto the conductor surface is uniquely
determined or not. One can say that maybe the charge can take another form
of distribution on the surface causing another potential for the
equipotential surface of the conductor and according to the theorem we
shall have another unique function for the potential outside the conductor.
In a geometric illustration there is not anything to prevent the above
problem for a sharp conductor being solved with equipotential surfaces
concentrated near either the sharp end or the other end; the charge is
concentrated at the sharp end in the first and at the other end in the
second case. Which occurs really is a matter that must be determined by
another uniqueness theorem, uniqueness theorem of charge distribution,
which has no relation to the uniqueness theorem of potential.
Analytical proof of this theorem is a problem that must be solved. That
this theorem is valid can be understood by some thinking and visualizing.
Separate from inner parts of the conductors consider external surfaces
of the conductors as some conducting thin shells. Obviously if some
charge is to distribute itself in these shells, the components of the
charge, as a result of the repulsive forces, will take the most distant
possible distances from one another, and even when for instance uncharged
conducting shells are set in the vicinity of charged conducting shells,
their conducting (or valence) charges will be separated in order that
like charges take the most distant and unlike charges take the most
neighboring possible distances from one another. What is clear is that
these "most"s indicate to some unique situation. Therefore we can say
that form of the surface charge distribution is a function of geometrical
form of the conductors and then will be specified uniquely for a definite
configuration of conductors.
II.B. Propotion of charge density to net charge
-----------------------------------------------
Now suppose that for a particular configuration of and definite amount of
charge given to some conductors we can find two distributions of charge in
the conductors in each of which the resultant electrostatic force on each
infinitesimal partial charge due to other infinitesimal partial charges
is outward normal to the conductor surface and there exists no tangential
component for this force. (Of course these outward normal forces are
balanced by surface stress in the material of the conductors.) Because
there is not any tangential component for the mentioned forces, existence
of these two charge distributions is possible. But because of the same
configuration for the both, the uniqueness theorem of charge distribution
necessitates that the both distribution be the same. We shall benefit form
this matter soon.
We prove that in a constant configuration of some conductors from which
only one has net charge, Q, change of this net charge form Q to <lambda>Q
causes that the surface density in each point of the conductors' surfaces
becomes <lambda>fold: Visualize the constant situation existent before that
Q becomes <lambda>Q. The charges in the conductors have a unique
distribution according to the uniqueness theorem of charge distribution.
In this distribution there exists a resultant electrostatic force exerted
on each infinitesimal partial surface charge <sigma>da due to other
partial charges which is outward normal to the conductor surface. Suppose
that this distribution becomes nailed up in some manner, ie each partial
charge becomes fixed in its position and no longer has the state of a
conducting free charge (in order that won't probably change its position as
a result of change of the charge). Now suppose each partial charge becomes
<lambda>fold in its position, ie we have for the new partial charge
<sigma>'da=<lambda><sigma>da. Since the partial charges are nailed up,
they are not free to redistribute themselves on the conductors' surfaces
probably. It is obvious that resultant electrostatic force exerted on a
partial charge <sigma>'da will be still outward normal to the conductor
surface, since firstly this partial charge is <lambda>fold of previous
<sigma>da and secondly each of other partial charges is <lambda>fold of
previous partial charges and then the only change in the resultant force
on <sigma>da will be in its magnitude which becomes <lambda>{2}fold,
while its direction will remain unchanged. Therefore, by changing each
<sigma>da to <lambda><sigma>da we have found a nailed up distribution
for the charges which exerts a resultant force on each partial surface
charge outward normal to the conductor surface, and furthermore, the only
change in the net charges of the conductors is in the conductor bearing
net charge Q previously which now bears <lambda>Q, and then it is obvious
that if the partial charges get free from the nailed state will remain
this distribution. Therefore, this distribution is a possible one, and
according to what said previously based on the uniqueness theorem, is the
same distribution that really occurs on the conductors' surfaces when the
net charge of the mentioned conductor changes from Q to <lambda>Q.
II.C. Generalization of the uniqueness theorem and of the charge density
------------------------------------------------------------------------
proportion to net charge
------------------------
In fact, the uniqueness theorem of charge distribution on the conductors
is true in case of a particular configuration of conductors and a constant
(nailed up) charge distribution and a constant set of linear dielectrics
in the space exterior to the conductors, ie in such a case a charge given
to the conductors causes a unique charge distribution on their surfaces.
The truth of this theorem can be found out with some indications similar
to previous ones.
Now consider a constant configuration of conductors and a constant set of
linear dielectrics outside the conductors. There is no charge outside the
conductors. We give a net charge to only one of the conductors. Certainly,
according to the above theorem we shall have a unique charge distribution
in the conductors. Suppose that the given charge of that conductor becomes
<lambda>fold. We want to prove that the surface free charge densties on
all of the conductors and also the dielectrics' polarizations will
become <lambda>fold consequently.
Visualize the situation existent before that the given charge becomes
<lambda>fold. An outward resultant force normal to the conductor surface
is exerted on each partial surface charge <sigma>da due to other
nonpolarization and polarization partial charges. Now suppose that all
the nonpolarization (or free) partial charges be nailed up in their
positions and then all the nonpolarization and polarization partial
charges (ie the previous free charges and dielectrics' polarizations)
become <lambda>fold. Obviously, in this case the resultant electrostatic
force on each partial surface charge is outward normal to the conductor
surface too (and only its magnitude has become <lambda>{2}fold).
Furthermore, it is obvious that in each point of each dielectric the
electrostatic field has only become <lambda>fold (without any change
in its direction) and then we see that this field is propotional to the
polarization at that point as must be so expectedly. Thus, if the charges
get free from the nailed state, they will remain on their positions,
and furthermore, the only change in net charges is in the above mentioned
conductor, net charge of which has now become <lambda>fold. Therefore,
this is a possible distribution and according to the above mentioned
uniqueness theorem of charge distribution is unique and then is the
same distribution that really occurs.
II.D. Superposition principle for the charge densities
------------------------------------------------------
We must also notice another point. We understood that in a configuration of
some conductors that only one of them has net charge, charge distribution
is unique. Suppose that we have N conductors and only conductor i has net
charge (Q[i]). The unique distribution that charges get, prescribes charge
surface density <sigma>[i](~r) (and polarization ~P[i](~r)) for each point
of each conductor (and each point outside the conductors).
Now consider this same configuration of these conductors from which only
conductor j (such that j is not equal to i) has net charge (Q[j]). The
unique distribution that charges get, prescribes charge surface density
<sigma>[j](~r) (and polarization ~P[j](~r)) for each point of each conductor
(and each point outside the conductors).
It is clear intuitively that if we have this same configuration of the
conductors from which only two conductors have net charges, the ith
conductor has the same relevant net charge (Q[i]) and the jth conductor
has the same relevant net charge (Q[j]), then the unique distribution that
charges get, prescribes charge surface density <sigma>[i](~r)+<sigma>[j](~r)
(and polarization ~P[i](~r)+~P[j](~r)) for each point of each conductor
(and each point outside the conductors). This fact has generality for
when each conductor has a specified net charge or when there is a fixed
distribution of external charge outside the conductors (ie we can add
contribution of this distribution towards forming charge surface density
on the conductors (and forming polarization) to other contributions).
We can even, when there are linear dielectrics, obtain surface charge
distribution on the conductors by adding the charge surface density in
each point on the conductors related to charge distribution in the
absence of dielectrics to the charge surface density in the same point
produced only by the polarizations of the dielectrics assuming that
there exists no net charge in any conductor but only the polarizations
exist.
Therefore, considering the theorems we have proved so far, we can
conceive that in a system of some charged conductors and some fixed
external charge distribution and some linear dielectrics if the net charge
of a conductor becomes <lambda>fold, free partial charge surface density
arising from that conductor, assuming that other conductors are uncharged
and there are not any dielectrics or other external charges, will become
<lambda>fold in each point on the conductors. It is evident that,
considering the integral definition of electrostatic potential and
assuming that the potential is zero at infinity, the partial potential
arising from that conductor (ie in fact from its effect on forming the
free charges) will become <lambda>fold in each point, too, and then the
partial potential arising from that conductor will become <lambda>fold
in each conductor which is an equipotential region for this partial
potential. In other words, the free net charge of one of the conductors
is proportional to the partial potential arising from the (effect of the
free net) charge of that conductor (assuming that there are not any
dielectrics or other external charges and that other conductors are
uncharged) in each of the conductors:
(i=1,2,3,...,N) <phi>{(j)}[i]=p[ij]Q[j]. Furthermore, this fact that
each conductor is an equipotential region for this partial potential
proves that p[ij] depends only on the geometry of the configuration
of the conductors and even does not depend on the dielectrics and their
positions (or other external charge distributions outside the conductors),
because, as we mentioned, this constant coefficient of the proportion,
p[ij], is related to when we suppose that there are not at all any
dielectrics (or other external charges) and infer that the charge
surface densities will become <lambda>fold if the net charge of a
conductor (the jth one) becomes <lambda>fold (assuming that other
conductors are uncharged).
Now since the potential of each conductor is the sum of its partial
potentials plus a constant, we have
<phi>[i]=<summation from j=1 to N>p[ij]Q[j]+c. (Adding of c removes the
worry arising from generalization of the necessity of the above reasoning
that the partial potentials must be zero at infinity.)
III. Static potential energy and current mistakes
-------------------------------------------------
III.A. Static potential energy
------------------------------
We know that if a closed surface S contains external electric charge Q and
polarization electric charge Q[P], then we shall have
<circulation over S>~E.^nda=(Q+Q[P])/<epsilon>[0]. In this relation ~E
is the partial electrostatic field arising from both an elective
distribuition of external charge, the part of which inside the closed
surface being equal to Q, and an elective distribution of polarization
charge, the part of which inside the closed surface being equal to
Q[P]. (The word "elective" implies that the entire existent charge
distribution is not necessarily taken into consideration, and similarly
the word "partial" implies that maybe only a part of the existent field
is intended. Notice the superposition principle of field and the
linearity of potential.)
On the other hand we have
Q[P]=<integral over S'>~P.^nda+<integral over V>(-<del>.~P)dv in which V
is the volume of the dielectric enclosed by S, and S' is the surface of
the conductors inside the closed surface S. In this relation ~P.^n and
-<del>.~P are the the polarization charge densities of the elective
distribution of polarization charge, and then we can say that in this
relation ~P is an elective (ie not necessarily entire) distribution of
electrostatic polarization. If by using the divergence theorem we change
the volume integral into the surface integral, we finally shall obtain
Q[P]=-<circulation over S>~P.^nda. The comparison of this relation with
the first relation of this section shows that
<circulation over S>(<epsilon>[0]~E+~P).^nda=Q in which ~P is an elective
distribution of polarization, and Q is the total charge of that part of
the elective distribution of external charge which is inside the closed
surface S, and ~E is the partial field arising from both the totality of
the elective distribution of external charge and the totality of the
elective distribution of polarization. On definition, the electric
displacement vector is ~D=<epsilon>[0]~E+~P. Then
<circulation over S>~D.^nda=<integral over V><rho>dv. This relation says
that if ~E is arising from both <rho>, which is an elective distribution
of external electric charge, and ~P, which is an elective distribution
of electrostatic polarization, then the surface integration of
~D=<epsilon>[0]~E+~P on the closed surface S is equal to the totality
of only that part of our elective external charge which is inside the
closed surface. If we use the divergence theorem in the recent relation,
we shall conclude <del>.~D=<rho>.
The electrostatic potential energy of a bounded system of electric
charges (which can exist in various forms of external charge, polarization
charge, etc, eg in the form of canceled charges, from the macroscopic
viewpoint, in a molecule) having the density <rho>, which is in fact the
spent energy for assembling all the fractions of the charge differentially
from infinity, is
U=1/2<integral over V[h]><rho>(~r)<phi>(~r)dv (1)
in which V[h] is the whole space and <phi> is the partial electrostatic
potential due to the distribution of <rho>. The way of obtaining the
relation (1) can be seen in many of the electromagnetic texts.
As it is so actually in the tridimentional world of matter, we disburden
ourselves from the dualizing the charge density as the surface and volume
ones and say we have only the volume density of the electrostatic charge
that, for instance, can have an excessive absolute amount on the surface
of a charged electric conductor. Now we take into consideration an elective
distribution of the volume density of the external (ie nonpolarization)
electric charge, <rho>. We want to obtain the electrostatic potential
energy of this distribution. We know that <del>.~D=<rho> so that
~D=<epsilon>[0]~E+~P in which ~P is the elective distribution of the
electrostatic polarization and ~E is the resultant field arising from
both the elective distribution of the external electric charge density
(<rho>) and the polarization charge densities due to the elective
distribution of the electrostatic polarization (~P). Since the
electrostatic polarization energy of this elective distribution of the
external electric charge is U=1/2<integral over V[h]><rho><phi>dv,
in which (V[h] is the whole space and) <phi> is only arising from <rho>
(not from both <rho> and ~P), then we shall have
U=1/2<integral over V[h]><phi><del>.~Ddv, and since
<integral over V[h]><phi><del>.~Ddv=
<integral over V[h]><del>.(<phi>~D)dv-<integral over V[h]>~D.<del><phi>dv=
<integral over S[h]><phi>~D.^n'da-<integral over V[h]>~D.<del><phi>dv=
0-<integral over V[h]>~D.(-~E[<rho>])dv=<integral over V[h]>~D.~E[<rho>]dv
(V[h] and S[h] being in turn the whole space and the total surfaces of
the problem (which of course there is not any surface)), then
U=1/2<integral over V[h]>~D.~E[<rho>]dv (2)
in which as we said " U is the electrostatic potential energy of an
elective distribution of the external electric charge with the density
<rho>, and we have <del>.~D=<rho> in which ~D=<epsilon>[0]~E+~P in which
~P is an elective distribution of electrostatic polarization and ~E is
arising from both ~P and <rho>, while ~E[<rho>] is the field arising only
from <rho>." It is obvious that this electrostatic potential energy has
been distributed in the space with the volume density u=1/2~D.~E[<rho>].
It is very opportune to compare the above accurate definition of the
electrostatic potential energy with what is set forth for discussion under
this very title in the present electromagnetic books, and to pay attention
to the existent inaccuracy in the definitions of the involved terms
caused by the omission of the subscript <rho> from the term ~E[<rho>].
This is a sample of the existent inaccuracies in the present current
electromagnetic theory specially in not correct distinguishing between
different electric fields. This mistake has caused that, considering
relation ~D=<epsilon>~E for linear dielectrics, wrong relations like
u=1/2<epsilon>E{2}=1/2D{2}/<epsilon> to be current in present
electromagnetic textbooks. We shall pay to some other mistakes soon.
III.B. Independence of capacitance from dielectric
--------------------------------------------------
Consider a system consisting of some fixed perfect conductors and some
linear dielectrics in the space exterior to the conductors and some fixed
distribution of external charge density in this space. We want to obtain
electrostatic potential energy arising from all the free net charges on
these conductors, ie the electrostatic potential energy of that part of
the charge distribution in all of the conductors which comes into
existence as a result of these free net charges (which of course does not
include electrostatic potential energy of the polarization and
distribution of external charges and that (other) part of the charge
distribution in all of the conductors which comes into existence as a
result of these polarization and external charges). Since each conductor
is an equipotential region for the potential arising from these free
net charges, for this electrostatic potential energy we have
U=1/2<summation from j=1 to N>Q[j]<phi>[j] from the relation (1), in which
Q[j] is the net charge of the conductor j and <phi>[j] is the electrostatic
potential on the conductor j arising from all free net charges of the
conductors of the system (ie one related to free net charges themselves
and their effect on the conductors, not also related to dielectric
polarization and other external charges and their effect on the conductors).
What is necessary to be emphasized again (and is important in the coming
discussion) is that the <phi>[j]'s are arising only from net charges of
the conductors not also from the polarization charges.
Using the coefficients of potential for this system we can also write
<phi>[i]=<summation from j=1 to N>p[ij]Q[j] in which Q[j] is the net
charge of the conductor j, and <phi>[i] is the electrostatic potential
on the conductor i arising from all (Q[j]'s ie all) net charges of the
conductors of the system (ie one related to free net charges themselves
and their effect in the conductors, not also related to dielectric
polarization and other external charges and their effect on the conductors).
Combining the two recent relations yields
U=1/2<summation from i=1 to N><summation from j=1 to N>p[ij]Q[i]Q[j]
for the electrostatic potential energy arising from free net charges of
the conductors of a system consisting of some perfect conductors and
probably some linear dielectrics and external charge distribution
outside the conductors.
A capacitor is defined as two conductors (denoted by 1 and 2) from
among a definite configuration of some conductors that one of them bears
net charge Q (Q being greater than or equal to zero), and the other one
bears -Q. (Existence of net charges on other conductors in the
configuration or of linear dielectrics or external charges outside the
conductors and the effect which each has on these two conductors
(ie 1 and 2) are not important at all. We shall find out this soon.)
Using the relation <phi>[i]=<summation from j=1 to N>p[ij]Q[j] for the
above capacitor we have:
<phi>[1]=p[11]Q+p[12](-Q)+0 )
> ===>
<phi>[2]=p[21]Q+p[22](-Q)+0 )
<cap. delta><phi>=<phi>[1]-<phi>[2]=(p[11]+p[22]-2p[12])Q=Q/C
(We know that p[12]=p[21] proof of which can be seen in many of the
electromagnetic books.) We have attention that in the relation
<cap. delta><phi>=Q/C, <cap. delta><phi> is the potential difference
between the potential arising from net charges of the conductors 1 and 2
(related to themselves and their effect in other conductors) on the
conductor 1 and the potential arising from these charges (related to
themselves and their effect in other conductors) on the conductor 2.
Therefore, since the potential of other charges is not considered and
considering linearity of potentials and that C, which is called as the
capacitance of the capacitor, depends only on the form of the configuration
of all (and not only two) of the conductors, then it is obvious that
existence of net charges on the conductors other than the conductors
1 and 2 and existence of any linear dielectrics or external charges in
the space exterior to the conductors, so far as the configuration of the
conductors is constant, are unimportant (and there is no need that one
of the conductors 1 and 2 be shielded by the other, the way presented
in some electromagnetic books for the potential difference independence
of whether other conductors are charged). We specially emphasize again that
so, we have proved that the capacitance (C) of a capacitor does not depend
on whether there exist any dielectrics at all and only depends on the
configuration of the conductors introduced in the definition of the
capacitor.
Using the relation
U=1/2<summation from i=1 to N><summation from j=1 to N>p[ij]Q[i]Q[j]
we obtain
U=1/2Q{2}/C=1/2Q<cap. delta><phi>=1/2C(<cap. delta><phi>){2} (3)
for the electrostatic potential energy of the charges Q and -Q (themselves
and of their effect). We should emphasize again that in the recent
relation, <cap. delta><phi> is the potential difference arising from
the free charges Q and -Q (and not also from eg polarization charges),
and C depends only on the configuration of the conductors (and not also
on eg existence or nonexistence of linear dielectrics).
At the end of this section let's obtain the capacitance of a capacitor
consisting of two parallel plates in which the plates separation d is
very small compared with the dimensions of the plates:
Q Q Q Q
C=----------------------=------=---------------------=-------------------
(<cap. delta><phi>)[Q] E[Q]d <sigma>d/<epsilon>[0] (Q/A)d/<epsilon>[0]
<epsilon>[0]A
=-------------- ,
d
in which (<cap. delta><phi>)[Q] and E[Q] are the potential difference and
the electrostatic field arising from Q and -Q (and not also from the
polarization charges) respectively. Therefore, the capacitance of this
capacitor is <epsilon>[0]A/d regardless of whether there exist any
linear dielectrics between the parallel plates or not.
And now see the present books of Electricity and Magnetism in which without
attention to this fact that <cap. delta><phi> must be arising only from the
capacitor charge, the relation <cap. delta><phi>=Ed, in which E is arising
from not only the capacitor charge but also the linear dielectrics
polarization charges, is used and consequently wrong expression
<epsilon>A/d is obtained for the capacitance.
III.C. Dielectric as source of potential
----------------------------------------
We saw that the mathematical discussions presented so far proved
independence of the capacitance of a capacitor from its dielectric.
But this is doubtlessly surprising for the physicists and engineers,
because they know well that dielectric has a substantial part in
accumulation of charge in the capacitor. This section is intended for
obviating this surprise.
It is made use often of electroscope to show the effect of dielectrics
in capacitors. If the two conductors of a charged capacitor are connected
to an electroscope, leaves of the electroscope will get away from each
other. Now, if, without any change in the configuration of the capacitor's
conductors, a dielectric is inserted between the two conductors of the
capacitor, the leaves of the electroscope will come close to each other.
Current justification of this phenomenon is as follows (eg see University
Physics by Sears, Zemansky and Young, Addison-Wesley 1987):
"The equation C=Q/<cap. delta><phi> shows the relation among the capacitor's
capacitance, capacitor's charge, and the potential difference between the
two conductors of the capacitor. When a dielectric is inserted into the
capacitor, due to the orientation of the electric dipoles of the dielectric
in the field inside the capacitor some polarization charge opposite to the
charge of each conductor of the capacitor is induced on that surface of
the dielectric which is adjacent to this conductor, and then the
electrostatic field in the dielectric, and thereby the potential difference
(between the two conductors), arising from both the capacitor's charge and
this induced polarization charge is decreased. Then, the denominator of
C=Q/<cap. delta><phi> decreases which results in increasing of the
capacitance (C) considering that Q remains uncharged, ie the capacitor's
capacitance increases by inserting a dielectric between the capacitor's
conductors. That the leaves of the electroscope come closer to each other
by inserting the dielectric is because of this same decreasing of the
potential difference, <cap. delta><phi>."
It is clear that considering the discussion presented in this article,
the above justification is quite wrong, because <cap. delta><phi> is the
potential difference arising only from the capacitor's charge not also
from the polarization charge formed in the dielectric. But why do the
leaves of the electroscope come closer to each other when a dielectric
is inserted into the capacitor? Its reason is quite obvious. Metal
housing and the leaves connected to the metal knob of the electroscope,
themselves, are in fact a capacitor, which when are connected separately
to the two conductors of the capacitor under measurement, a new
(equivalent) capacitor will be formed consisting of two conductors: the
first being one of the conductors of the capacitor under measurement
and the electroscope's metal housing which is connected to it, and the
second being the other conductor of the capacitor under measurement and
the set of the knob and the leaves of the electroscope which is connected
to this conductor. It is obvious that if the capacitor under measurement
is charged at first, its charge now, after its connecting to the
electroscope, will be distributed throughout the new formed capacitor
and then a part of the charge of the primary capacitor now will go to
the electroscope because of which the leaves of the electroscope will get
away from each other (because the opposite charges induced in the
electroscope will attract each other causing drawing of the leaves toward
the electroscope's housing which itself means more separation of the
leaves from each other).
Inserting the dielectric into the capacitor we cause creation of
polarization charges in the dielectric which this, in turn, causes more
charges of the new formed capacitor to be drawn towards the dielectric.
Thus, the distribution of the charge will be changed in such a manner
that a part of the charge distribution in the electroscope will go to
the primary capacitor (or the one under measurement) to be placed as close
as possible to the dielectric; this means decrease of the electroscope's
charge which will cause its leaves to come closer to each other.
Therefore, the act of the dielectric is change of the charge distribution
in the new capacitor formed from the primary capacitor and the electroscope,
not change of the capacitance of the primary capacitor.
Now, let's connect the two plates of a parallel-plate capacitor by a wire
in the space exterior to the space between the plates. What will happen if a
slice of a dielectric having a permanent electric polarization is inserted
between the two plates of the capacitor? The polarized dielectric will cause
induction of charge on the two plates; the positive surface of the slice will
induce negative charge on the plate adjacent to it, and the negative
surface will induce positive charge on the (other) plate adjacent to it.
Induction of charge on the two plates, while they had no charge
beforehand, means that while inserting the dielectric between the plates
an electric current has been flowing in the wire from one plate to the
other. In other words the dielectric acts like a power supply producing
electric current or charging the capacitor. Then, we can attribute
electric potential difference to it (like the potential difference
between the two poles of a battery).
Now, how will the situation be if the inserted dielectric is not to have
previous polarization but it is to be polarized because of the charge
(or in fact the electric field produced by the charge) of the capacitor?
Answer is that the situation will be similar to the same state of
permanent polarization, and again the dielectric acts as a source of
potential. Its physical and direct reason can be seen easily in the
discussion we presented about the electroscope. There, we saw that
inserting the dielectric, charge distribution was changed in such a manner
that some more charges were accumulated on the conductors of the (primary)
capacitor. It is clear that more accumulation of charge on the capacitor
necessitates flowing of electric current in the circuit. Cause of this
current and of the more accumulation of charge on the capacitor is the
source of potential difference which we must attribute to the dielectric.
In this manner, the purpose of this section has been fulfilled practically;
in electric circuits wherever a dielectric is to exist between the
conductors of a capacitor, a proper source of voltage must be considered
in the circuit in the same place of the dielectric. Such a voltage source
causes accumulation of charges on the conductors of the capacitor more
than when there exists no dielectric in the capacitor. One can say
whether this act is not equivalent to defining, in principle, the
capacitance of a capacitor equal to the accumulated charge on the capacitor
(due to both the configuration of the capacitor's conductors and the
electric induction in the conductors caused by the polarization of the
dielectric) divided by the potential difference between the two conductors
of the capacitor (which is the method that current instruments measuring
capacitor's capacitances work based on it) and no longer considering the
dielectric as a source of potential. Following example shows that
consequences of such a definition in practice are not equivalent to the
practical consequences of the main definition of capacitance of capacitor
(although can be close to it under suitable conditions). We then shall
investigate another example which will show, well, considerable differences
that can come into existence if role of the dielectric as a power supply
in the circuit is not taken into consideration, according to which a quite
practical criterion for testing the theory presented in this section in
comparison with the current theory will be presented.
III.D. Some examples as test
----------------------------
Let's connect the two plates of a dielectricless parallel-plate capacitor
to the two poles of a battery. At the end of the section III.B. we saw that
the capacitance of such a capacitor is <epsilon>[0]A/d in which A the
capacitor's area and d is the distance between its plates. Then, according
to the relation C=Q/<cap. delta><phi> for the capacitor's capacitance, we
have <epsilon>[0]A/d=<sigma>A/V in which <sigma> is the surface density
of the charge accumulated on the capacitor and V is the potential difference
given to the two plates of the capacitor by the battery. In this manner
we have:
<sigma>d=<epsilon>[0]V (4)
Now we fill the space between the two plates with a linear dielectric with
the permitivity <epsilon>. We indicate the magnitude of the formed electric
polarization in the dielectric by P. P is in fact equal to the surface
density of the polarization charge in the dielectric. Suppose that a charge
exactly equal to the polarization charge is induced on the plates of the
capacitor. (Indeed, in the state of induction of charge in the capacitor
due to the polarized dielectric between the capacitor's plates we should
suppose that the two plates of the capacitor are connected to each other
by a wire in the space exterior to the space between the plates; in other
words in this state the battery existent in the circuit does not play
any role except as a short circuit.) Then the charge induced on the
capacitor due to the polarization of the dielectric is equal to PA. This
charge, as we said, has been stored in the capacitor because of a source
of potential difference, equal to V', which we must attribute to the
dielectric; ie because of the potential difference V' exerted to the two
plates of the capacitor the charge PA has been accumulated in the
capacitor, and then the ratio PA/V' is equal to the capacitor's capacitance
<epsilon>[0]A/d=PA/V'. Considering that P=(<epsilon>-<epsilon>[0])E=
(<epsilon>-<epsilon>[0])<sigma>/<epsilon> in which E is the electrostatic
field arising from both the external and polarization charges we infer
from this relation that
V'=(<epsilon>-<epsilon>[0])<sigma>d/(<epsilon><epsilon>[0]) which
considering Eq.(4) results in
V'=(1-<epsilon>[0]/<epsilon>)V. (5)
Let's calculate sum of the charges (Q) accumulated on this capacitor
(due to both the configuration of the capacitor's conductors and the
induction arising from the (polarization of the) dielectric). For this
act we must add the potential difference arising from the dielectric to
the potential difference given by the battery and then multiply the sum
by the (real) capacitance of the capacitor <epsilon>[0]A/d:
Q=(V+(1-<epsilon>[0]/<epsilon>)V)<epsilon>[0]A/d=
(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d V (6)
Can we present another definition of capacitance of capacitor, for
convenience in practice, equal to sum of the accumulated charges on the
capacitor (consisting of the charges arising from both the configuration
of the capacitor's conductors and the induction due to the dielectric)
divided by the potential difference between the two capacitor's conductors,
given to the capacitor only by the battery (or the circuit)? Considering
Eq.(6) such a definition gives this (newly defined) capacitance of our
capacitor equal to
Q/V=(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d. (7)
Is this definition useful in practice, and does it yield real consequences?
The answer is negative. It is sufficient only instead of a single
capacitor to consider n capacitors connected in series such that the
space between the plates of only one of them is filled with dielectric
and to try to calculate the accumulated charges on the equivalent capacitor.
If all of these n capacitors were dielectricless, because of the identity
between the capacitors the (shared) potential difference between the
two plates of each of these capacitors would be V/n. When only one of
these capacitors is filled with a linear dielectric with the permittivity
<epsilon>, the potential difference related to this dielectric (as a
source of potential), similar to Eq.(5) will be
(1-<epsilon>[0]/<epsilon>)V/n. Since these n capacitors are identical and
the capacitance of each of them is <epsilon>[0]A/d, the equivalent
capacitance of these n capacitors which are connected in series will be
obtained by solving the equation 1/C[1]=n/(<epsilon>[0]A/d) for C[1]
equal to <epsilon>[0]A/(nd). Therefore, the charge accumulated on each
capacitor is equal to
<epsilon>[0] V <epsilon>[0]A
( V + ( 1 - ------------ ) --- ) -------------
<epsilon> n nd
<epsilon>-<epsilon>[0] <epsilon>[0]A
= ( 1 + ---------------------- ) ------------- V . (8)
n<epsilon> nd
But now let's see if the capacitance of the capacitor having dielectric is
to be equal to (7) while the capacitance of each of the other capacitors is
equal to <epsilon>[0]A/d, whether or not the charge accumulated on each
capacitor will be obtained still equal to (8) when no longer the source of
potential difference related to the dielectric is considered in lieu of
considering (7) for the capacitance of the capacitor having dielectric.
Equivalent capacitance of the capacitors which are in series will be
obtained by solving the equation
1 n-1 1
---- = --------------- + -----------------------------------------
C[2] <epsilon>[0]A/d (2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d
for C[2], and charge of each capacitor should be considered equal to C[2]V:
1 <epsilon>[0]A
C[2]V= --------------------------------------- . ------------- V (9)
n-1+<epsilon>/(2<epsilon>-<epsilon>[0]) d
Obviously the coefficient of <epsilon>[0]AV/d in Eq.(8) is not equal to
the coefficient of <epsilon>[0]AV/d in Eq.(9) except when
<epsilon>=<epsilon>[0] or n=1. Thus, we see that the new definition we
tried to present for capacitance of capacitor is not so useful in practice
(at least in this example does not give the real charge accumulated on
the capacitors). But the ratio of these two coefficients is not so far from
one. To see this fact let's indicate <epsilon>/<epsilon>[0] by K and
obtain the ratio of the coefficient of <epsilon>[0]AV/d in Eq.(9) to the
coefficient of <epsilon>[0]AV/d in Eq.(8):
(n-1+<epsilon>/(2<epsilon>-<epsilon>[0])){-1} (K-1){2}(n-1)
---------------------------------------------- = 1/(1 + -------------)
1/n + (<epsilon>-<epsilon>[0])/(n{2}<epsilon>) (2K-1)Kn{2}
It is seen that the degree of the term (K-1){2}(n-1)/((2K-1)Kn{2}) with
respect to K is zero and with respect to n is -1; then this term is close
to zero practically, or in other words the ratio of the above mentioned
coefficients is close to one practically. This matter is itself a good
reason that why the definition of capacitance in the form of capacitor's
charge divided by the potential difference exerted on the capacitor's
conductors (Eq.(7)) has been able to endure practically and the difficulties
due to such a definition has remained hidden in practice. But, important
for a physicist should be mathematical much exactness and discovery of
what actually occurs or exists. In order to find out that such an
exactness can be important even in practice (and then won't be negligible
even for engineers) notice the following example.
Consider a series circuit of RLC, which its capacitor is parallel-plate
and dielectricless, connected to a constant voltage V. After connection
of the switch in the time t=0, the equation of the circuit will be
V=RI+LdI/dt+1/(2C)<integral from t=0 to t>I(t)dt. (10)
(We should notice that as it will be proved in the last section of this
article, in this circuit we must consider the circuital potential
difference of the capacitor, ie the third term of the right-hand side
of (10), not as it is usual wrongly its electrostatic potential difference
ie 1/C<integral from t=0 to t>I(t)dt. Another noticeable point being that
as it has been explained in the section 5 of the 13th article of the book,
L in (10) is in fact equal to <mu><epsilon>'a'L[B]{*} not equal to only
d<cap. phi>[B]{*}/dI(=L[B]{*}) according to its usual definition.) With one
time differentiation of this equation with respect to time, the following
equation will be obtained considering that V is constant:
Ld{2}I/dt{2}+RdI/dt+I/(2C)=0. If R/(2L)<(2LC){-1/2}, this equation will be
solved as I= a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>) in which
1 R{2}
<omega>[n] = (--- - -----){1/2} (11)
2LC 4L{2}
and a and <theta> are two arbitrary constants. Since in t=0 we have I=0
and then also from Eq.(10) we have dI/dt=V/L, then a=V/(<omega>[n]L) and
<theta>=<pi>/2, and then
V
I= ---------- exp(-Rt/(2L))sin(<omega>[n]t). (12)
<omega>[n]L
For calculating the voltage drop in the capacitor we should calculate the
third term of the right-hand side of Eq.(10):
1 V
--<integral from t=0 to to t>-----------exp(-Rt/(2L))sin(<omega>[n]t)dt =
2C <omega>[n]L
R
V(1-exp(-Rt/(2L))(cos(<omega>[n]t)+------------sin(<omega>[n]t))) (13)
2<omega>[n]L
Now, if the space between the two plates of the capacitor (without any
change in the configuration of the plates) is to be filled by a linear
dielectric with the permittivity <epsilon>, we must multiply the negative
of the voltage drop in the capacitor ((13)) by (1-<epsilon>[0]/<epsilon>)
till according to Eq.(5) the potential difference which we must attribute
to the dielectric as source of potential is otained. We then should add
this source to the previous constant source and equate the sum to the
right-hand side of Eq.(10):
V+V(exp(-Rt/(2L))(cos(<omega>[n]t)+R/(2<omega>[n]L)sin(<omega>[n]t))-1)(1-<ep
silon>[0]/<epsilon>)=RI+LdI/dt+1/(2C)<integral from t=0 to t>I(t)dt (14)
With one time differentiation of this equation with respect to time the
following equation will be obtained:
d{2}I dI 1 <epsilon>[0] 2<omega>[n]L
L----- +R-- + --I=V(1- ------------)------------exp(-Rt/(2L))sin(<omega>[n]t)
dt{2} dt 2C <epsilon> R{2}C-2L
Particular solution of this equation is
V <epsilon>[0]
--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t),
2L-R{2}C <epsilon>
and general solution of its corresponding homogeneous equation is
a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>) with the two arbitrary constants
a and <theta>. Then general solution of this equation is
V <epsilon>[0]
I=a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>)+ --------(1- ------------)t.exp(
2L-R{2}C <epsilon>
-Rt/(2L))cos(<omega>[n]t)
with the two arbitrary constants a and <theta>. For obtaining a and <theta>
by means of the initial conditions, we should be careful that initial
conditions must be fit, ie t=0 should be the same moment that, without
dielectric, the current in the circuit was zero and we had dI/dt=V/L; and
now, when the dielectric has been inserted, we should see how the conditions
change, and in this moment (t=0) what the current and its time derivative
are as initial conditions. The physics of the problem says that we have
in this state I=0 in this moment too, and then also it is clear from Eq.(14)
that in this moment we have dI/dt=V/L too. Then
V V <epsilon>[0]
a = ----------- + --------------------(1- ------------) =
<omega>[n]L <omega>[n](R{2}C-2L) <epsilon>
L(1+<epsilon>[0]/<epsilon>)-R{2}C V
---------------------------------.----------- and <theta>=<pi>/2. Thus
2L-R{2}C <omega>[n]L
L(1+<epsilon>[0]/<epsilon>)-R{2}C V
I=---------------------------------.-----------exp(-Rt/(2L))sin(<omega>[n]t)+
2L-R{2}C <omega>[n]L
V <epsilon>[0]
--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t). (15)
2L-R{2}C <epsilon>
(It is noticeable that when <omega>=<omega>[0] the same Eq.(12) will be
obtained from this equation.) We obtained Eq.(15) for the current of the
circuit, while what is current at present is that inserting the linear
dielectric (with the permittivity <epsilon>) between the plates of the
capacitor only the capacitor's capacitance changes from C to KC where
K=<epsilon>/<epsilon>[0] (without any addition of new source of potential
to the circuit), and then the circuit's current has the same form of
Eq.(12) with this only difference that in the equation related to
<omega>[n] (Eq.(11)) we must write KC instead of C.
Now suppose that instead of the constant voltage V we have an alternating
voltage in the form of V(t)=V[0]sin(<omega>t-<theta>')(in which <theta>'
is a constant value) as the main source of potential in the series circuit
of RLC which its parallel-plate capacitor is dielectricless. In such a case
we have
dI 1
V[0]sin(<omega>t-<theta>')=RI+L-- + --<integral from t=0 to t>I(t)dt, (16)
dt 2C
and then Ld{2}I/dt{2}+RdI/dt+I/(2C)=V[0]<omega>cos(<omega>t-<theta>').
Particularl solution of this equation is
a[1]cos(<omega>-<theta>'-<thata>[1]) in which
a[1]=V[0]/((1/(2C<omega>)-L<omega>){2}+R{2}){1/2} (17)
and
<theta>[1]=cot{-1}((1/(2<omega>C)-L)/R) (18)
Since solution of its corresponding homogeneous equation is
a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>), the general siolution of this
equation is
I=a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>)+
a[1]cos(<omega>t-<theta>'-<theta>[1]) (19)
with the two arbitrary constant a and <theta> (of course assuming that
R/(2L)<(2LC){-1/2}).
We suppose that we have I=0 in t=0 and from Eq.(16) we have
dI/dt=-V[0]sin<theta>'/L in this moment. Having these initial values we can
obtain a and <theta>, but since the first term of the right-hand side of
Eq.(19) is transient, this act is of no importance for us.
Now, as before, having the form of current (Eq.(19)) we obtain voltage drop
in the capacitor:
1/(2C)<integral from t=0 to t>(a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>)+
a[1]cos(<omega>t-<theta>'-<theta>[1]))dt=a(exp(-Rt/(2L))(<omega>[n]Lsin(<ome
ga>[n]t-<theta>)-R/2cos(<omega>[n]t-<theta>))+<omega>[n]Lsin<theta>+R/2cos<
theta>)+a[1]/(2<omega>C)(sin(<omega>t-<theta>'-<theta>[1])+sin(<theta>'+<the
ta>[1])) (20)
And now, as before, if the space between the two plates of the capacitor
is to be filled by a linear dielectric with the permittivity <epsilon>
(without any change in the plates' configuration), in order to obtain the
potential difference that we must attribute to the dielectric as a source
of potential in the circuit, according to Eq.(5) we should multiply the
negative of the potential drop in the capacitor (20) by
(1-<epsilon>[0]/<epsilon>). We then must add this source with the initial
alternating source and equate the sum to the right-hand side of Eq.(16):
V[0]sin(<omega>t-<theta>')+a(1-<epsilon>[0]/<epsilon>)(exp(-Rt/(2L))(R/2cos(
<omega>[n]t-<theta>)-<omega>[n]Lsin(<omega>[n]t-<theta>))-<omega>[n]Lsin<the
ta>-R/2cos<theta>)-a[1]/(2<omega>C)(1-<epsilon>[0]/<epsilon>)(sin(<omega>t-
<theta>'-<theta>[1])+sin(<theta>'+<theta>[1]))=RI+LdI/dt+1/(2C)<integral
from t=0 to t>I(t)dt
With one time differentiation of this equation with respect to time the
following equation will be obtained:
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=V[0]<omega>cos(<omega>t-<theta>')-a(1-<epsilon>[
0]/<epsilon>)(R{2}/(4L)+<omega>[n]{2}L)exp(-Rt/(2L))cos(<omega>[n]t-<theta>)
-a[1]/(2C)(1-<epsilon>[0]/<epsilon>)cos(<omega>t-<theta>'-<theta>[1]) (21)
For obtaining the particular solution of this equation we must add up
particular solutions of the following equations (for reason see
Differential Equations with Application and Historical Notes by Simmons,
McGraw-Hill Inc., 1972):
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=V[0]<omega>cos(<omega>t-<theta>') (22)
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=-a(1-<epsilon>[0]/<epsilon>)(R{2}/(4L)+
<omega>[n]{2}L)exp(-Rt/(2L))cos(<omega>[n]t-<theta>) (23)
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=-a[1]/(2C)(1-<epsilon>[0]/<epsilon>)cos(
<omega>t-<theta>'-<theta>[1]) (24)
We then must add the obtained particular solution to the general solution
of the corresponding homogeneous equation to obtain the general solution
of Eq.(21).
Both the general solution of the homogeneous equation and particular
solution of Eq.(23) are (trigonometric) multiples of exp(-Rt/(2L)), thus
these two terms in the general solution of Eq.(21) are transient and then
unimportant for us. Then, for obtaining the nontransient part of the
general solution of Eq.(21) we should obtain the particular solution of
the equations (22) and (24) and then add them up.
Particular solution of Eq.(22) is
( 2V[0]C<omega>/(4L{2}C{2}<omega>{4}+4(R{2}C-L)C<omega>{2}+1) )((1-2LC<ome
ga>{2})cos(<0mega>t-<theta>')+2RC<omega>sin(<omega>t-<theta>'))
and particular solution of Eq.(24) is
( -a[1]/(4L{2}C{2}<omega>{4}+4(R{2}C-L)C<omega>{2}+1) )(1-<epsilon>[0]/<ep
silon>)((1-2LC<omega>{2})cos(<omega>t-<theta>'-<theta>[1])+2RC<omega>sin(
<omega>t-<theta>'-<theta>[1])).
If we write the trigonometric terms in the recent solution in terms of the
sine and cosine of the arguments (<omega>t-<theta>') and <theta>[1], and
add up the particular solutions obtained for the equations (22) and (24),
and equate the sum to the expression a[2]cos(<omega>t-<theta>'-<theta>[2]),
and use the equations (17) and (18), we shall finally obtain:
a[2]cos<theta>[2]=2V[0]C<omega>.((1-2LC<omega>{2})(4R{2}C{2}<omega>{2}+(1-
2LC<omega>{2}){2})+(1-<epsilon>[0]/<epsilon>)(4R{2}C{2}<omega>{2}-(1-
2LC<omega>{2}){2}))/(4R{2}C{2}<omega>{2}+(1-2LC<omega>{2}){2}){2} (25)
and
a[2]sin<theta>[2]=4V[0]RC{2}<omega>{2}.(4R{2}C{2}<omega>{2}+(1-2LC<ome
ga>{2}){2}-2(1-<epsilon>[0]/<epsilon>)(1-2LC<omega>{2}))/(4R{2}C{2}<ome
ga>{2}+(1-2LC<omega>{2}){2}){2} (26)
We can solve these equations to obtain a[2] and <theta>[2] till the
nontransient solution a[2]cos(<omega>t-<theta>'-<theta>[2]) for the
circuit current will be obtained unambiguously. The value which is
obtained for the amplitude a[2] from these equations is
2V[0]C<omega>(4R{2}C{2}<omega>{2}+(1/K-2LC<omega>{2}){2}){1/2}
a[2]=-------------------------------------------------------------- (27)
4R{2}C{2}<omega>{2}+(1-2LC<omega>{2}){2}
in whivh K=<epsilon>/<epsilon>[0]. (It is easily seen that for K=1 the
same amplitude a[1] presented in Eq.(17) will be obtained from a[2].)
Now if, as it is thought at present, after inserting the dielectric
between the capacitor's plates its capacitance is to increase to KC and
no more, then we must conclude that the amplitude of the (nontransient)
current is in the same form shown in Eq.(17) except that KC must be
substituted for C in this equation. Namely, the magnitude of such an
amplitude will be:
(V[0]/((1/(2K<omega>)-L<omega>){2}+R{2}){1/2}=) 2V[0]C<omega>/(4R{2}C{2}<ome
ga>{2}+(1/K-2LC<omega>{2}){2}){1/2} (=2V[0]C<omega>(4R{2}C{2}<omega>{2}+(1/K
-2LC<omega>{2}){2}){1/2}/(4R{2}C{2}<omega>{2}+(1/K-2LC<omega>{2}){2})) (28)
A comparison between (27) and (28) shows that their variations with K is
opposite to each other, ie if (27) increases with increase of K, (28) will
decrease with increase of K, and if (27) decreases with increase of K, (28)
will increase with increase of K, and vice versa. For example on condition
that <omega>{2} being greater than or equal to 1/(2LC) the expression (27)
indicates that the current's amplitude increases by inserting the
dielectric, while the expression (28) says that this amplitude must
decrease under the same condition. Investigating that whether or not
experiment shows that provided that <omega>{2} being greater than or equal
to 1/(2LC) current intensity of the circuit increases by inserting
dielectric between the capacitor's plates is a good test for accepting
the theory presented here and rejecting the current one or vice versa.
For finding the resonance frequency of the circuit it is sufficient to
differentiate from the right-hand side of Eq.(27) with respect to <omega>
and then to equate the obtained result to zero and to solve the obtained
equation for <omega>. By doing this act we obtain the following result
for the square of the resonance frequency <omega>[r]:
2(K-1)+(4(K-1){2}+1){1/2}
<omega>[r]{2}= ------------------------- (29)
2LCK
(It is seen that for K=1, square of the resonance frequency is 1/(2LC)
which is just the same result which Eq.(17) predicts for the square of
the resonance frequency. (Reminding of this point is necessary that as
we said we have L=<mu><epsilon>'a'L[B]{*} here.))
Now, let's see what the prediction of the present current belief is for
the resonance frequency of the circuit. It says that since inserting the
dielectric (according to its belief) the amplitude of the current is
V[0]/((1/(2KC<omega>)-L<omega>){2}+R{2}){1/2} (see Eq.(28)), then the
square of the resonance frequency will be:
1
---- (30)
2LCK
A simple mathematical try shows that the coefficient of 1/(2LC) in (29)
(ie (2(K-1)+(4(K-1){2}+1){1/2})/K) is an ascending function of K, while
the coefficient of 1/(2LC) in (30) (ie 1/K) is a descending function of K.
Namely, the analysis presented here shows that by inserting dielectric
between the capacitor's plates the resonance frequency increases, while
according to the current belief this frequency must decrease. That actually
whether or not the resonance frequency of the circuit increases with
inserting dielectric between the plates of the capacitor (without any
change in the plates' configuration) is a quite practical test for
establishing the validity of the theory presented in this article and
invalidity of the current belief in this respect, or vice versa.
III.E. Again parallel-plate capacitor as another test
-----------------------------------------------------
Now we obtain the electrostatic potential energy of the parallel-plate
capacitor mentioned at the end of the section III.B. by two methods.
First, using the relation U=1/2C(<cap. delta><phi>)[Q]{2} we obtain
U=1/2(<epsilon>[0]A/d)(<cap. delta><phi>)[Q]{2}.
In the second method we use the relation (2), ie U=1/2<integral over
V[h]>~D.~E[Q]dv in which ~E[Q] is the field arising from Q and -Q (and
not also from the polarization charges). We have the following relation:
~D=<epsilon>~E=<epsilon>(~E[Q]+~E[P])=<epsilon>~E[Q]+<epsilon>~E[P] (31)
in which ~E[P] is the field arising only from the polarization charges
of the dielectric set between the two plates. Let's obtain ~E[P] in terms
of ~D. Suppose that ~P is the polarization of the dielectric and ^n is the
unit vector in the direction of ~E. ~P.(-^n) is the polarization charge
surface density formed adjacent to the plate bearing the (positive) charge
Q, and ~P.^n is the polarization charge surface density formed adjacent to
the plate bearing the charge -Q. Since ~P=(<epsilon>-<epsilon>[0])~E then
~P.(-^n)=(<epsilon>[0]-<epsilon>)E and ~P.^n=(<epsilon>-<epsilon>[0])E
which the first is negative and the second is positive obviously. Then,
the electrostatic field arising from these (polarization) charges in the
dielectric is
~P.^n <epsilon>[0]-<epsilon>
~E[P] = ------------(-^n) = ----------------------~E (32)
<epsilon>[0] <epsilon>[0]
and since ~D=<epsilon>~E then
~E[P]=(<epsilon>[0]-<epsilon>)/(<epsilon>[0]<epsilon>)~D. Combining this
result with the relation (31) yields
<epsilon>[0]-<epsilon>
~D=<epsilon>~E[Q]+ ----------------------~D ==> ~D=<epsilon>[0]~E[Q]. (33)
<epsilon>[0]
Therefore, we have U=1/2<integral over V[h]>~D.~E[Q]dv=1/2<integral over
V=Ad><epsilon>[0]E[Q]{2}dv=1/2<epsilon>[0]AdE[Q]{2}=1/2<epsilon>[0]Ad((<cap.
delta><phi>)[Q]/d){2}=1/2(<epsilon>[0]A/d)(<cap. delta><phi>)[Q]{2}, which
is the same result obtained in the first method.
Now we proceed to another case. Consider the following figure.
x
<----------->
______________________________
---------------------- /^\
......................| |
......................| | d
______________________| \|/
-----------------------------"
<---------------------------->
l
Figure. A linear dielectric block is pulled into the space between the
plates of a parallel-plate capacitor having the constant
electrostatic potential difference (<cap. delta><phi>)[Q].
The (unshown) width of the plates is w. A linear dielectric block is along
the l-dimension and only the length x is between the plates. Potential
difference between the two plates is constant (equal to (<cap. delta><
phi>)[Q]; we proved this fact beforehand). It is clear that the charges
on that part of a plate of the capacitor which is in the empty part of
the capacitor exert an attractive force on the polarization charges
adjacent to that plate and a repulsive force on the polarization charges
adjacent to the other plate, while the charges on the empty part of the
other plate act a similar work, and the resultant force of all of these
forces is an inward force along the l-dimension magnitude of which must
approach zero when d approaches zero. Now let's try to obtain this force
from the energy method. First of all, according to what said so far, it is
obvious that with the dielectric displacement the electrostatic potential
energy of the capacitor being only of the capacitor charge (Q and -Q)
does not alter. Thus, only the electrostatic potential energy of the
dielectric and its alteration must be considered.
We know that the surface density of polarization charge of the dielectric
in the capacitor is +P or -P and then the electrostatic field arising from
it is ~E[P]=-~P/<epsilon>[0]. On the other hand, by using each of the
relations (1) and (2) we obtain a unique expression for the electrostatic
potential energy of only the polarization charges of the dielectric:
(1) ==> U[P]=1/2<integral over V[h]><rho><phi>dv=1/2((-Pd/(2<epsilon>[0])+
0)(Q[P])+(-(-Pd)/(2<epsilon>[0])+0)(-Q[P]))=Pd/(2<epsilon>[0])Q[P]=Pd/(2<ep
silon>[0])P(wx)=P{2}d/(2<epsilon>[0])wx
considering that the potential arising from an infinite charged plate with
the surface charge density <sigma> is -<sigma>/(2<epsilon>[0])d at the
(nonnegative) distance d from the plate, and
(2) ==> U[P]=1/2<integral over V[h]><epsilon>[0]~E[P].~E[P]dv=<epsilon>[
0]/2<integral over V[h]>~E[P]{2}dv=<epsilon>[0]/2<integral over V[h]>(P/<ep
silon>[0]){2}dv=<epsilon>[0]/2 P{2}/<epsilon>[0]{2} wxd=P{2}d/(2<epsilon>[
0])wx.
We have also ~P=-<epsilon>[0]~E[P] from ~E[P]=-~P/<epsilon>[0]. If in
addition we apply the relations (32), (33) and (31), we shall obtain
~P=-<epsilon>[0]~E[P]=(<epsilon>-<epsilon>[0])~E=(<epsilon>[0](<epsilon>-
<epsilon>[0])/<epsilon>)~E[Q] and consequently
U[P] = P{2}d/(2<epsilon>[0]) wx = <epsilon>[0]E[P]{2}d/2 wx = (<epsilon>-<ep
silon>[0]){2}E{2}d/(2<epsilon>[0]) wx = <epsilon>[0](<epsilon>-<epsilon>[
0]){2}E[Q]{2}d/(2<epsilon>{2}) wx.
Since with the displacement of the dielectric only x is changed, then
dU[P] = <epsilon>[0]E[P]{2}d/2 wdx = (<epsilon>-<epsilon>[0]){2}E{2}d/(2<ep
silon>[0]) wdx = <epsilon>[0](<epsilon>-<epsilon>[0]){2}E[Q]{2}d/(2<epsi
lon>{2}) wdx. (34)
We know that the above mentioned force pulling the dielectric into the
capacitor performs some work on the dielectric which, according to the
conservation law of energy, this work must be conserved in some manner.
By pulling inward, this force not only causes forming more polarization
charges, but also alters (and in fact increases) the kinetic energy of the
dielectric block. Thus, the above mentioned work is conserved both as the
electrostatic potential energy of the formed polarization charges and as
the alteration of the kinetic energy. We show this work as dW and the
alteration of the electrostatic potential energy as dU[P] and the
alteration of the kinetic energy as dT. Therefore, we have:
dW=dU[P]+dT & dW=F[x]dx ==> F[x]dx=dU[p]+dT )
> ==>
(34) )
F[x]dx = <epsilon>[0](<epsilon>-<epsilon>[0]){2}wd/(2<epsilon>{2}) E[Q]{2}dx
+ dT = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2}) w (<cap. del
ta><phi>)[Q]{2}/d dx + dT (35)
It is obvious that if in an especial case we have dT=0 then we shall have
F[x] = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2}) E[Q]{2}wd
= <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2}) w (<cap. delta><
phi>)[Q]{2}/d. (36)
(It is seen that as was predicted beforehand, this force will approach zero
if d approaches zero.)
Observing the present current mistakes (including what we saw about the
capacitance and electrostatic potential energy of a capacitor) we see
the following relation instead of Eq.(35) in the present books of
Electricity and Magnetism or Electromagnetism:
F[x]dx = 1/2 (<epsilon>-<epsilon>[0]) w (<cap. delta><phi>)[Q]{2}/d dx =
1/2 (K-1)<epsilon>[0]E[Q]{2}(wd)dx (37)
where it is supposed that (<cap. delta><phi>)[Q] remains constant (by
retaining the plates connected to the poles of a battery).
And also by mistake the following general result (instead of the especial
result (36)) is inferred from the relation (37):
F[x] = 1/2 (<epsilon>-<epsilon>[0]) w (<cap. delta><phi>)[Q]{2}/d = 1/2 (
K-1)<epsilon>[0]E[Q]{2}wd
Practical comparison of the above relations for testing the truth of
Eq.(35) experimentally should be possible preparing ideal conditions and
regarding fringing effects at the edges of the capacitor.
IV. Two kinds of potential difference for a capacitor
-----------------------------------------------------
At present in all the textbooks of Electricity and Magnetism wherever
electrostatic potential difference between the two conductors of a
capacitor is concerned if to its producer source, ie the battery, has
been pointed implicitly or explicitly, it is shown or stated implicitly
or explicitly that this electrostatic potential difference is equal to
the potential difference between the two poles of the battery that has
charged the capacitor. But we now shall prove easily that the
electrostatic potential difference between the two conductors of a
capacitor is twofold compared with the potential difference between the
two poles of the battery which has charged it.
Suppose that the potential difference between the two poles of the
battery is <cap. delta><phi> and the electrostatic potential difference
between the two conductors of the capacitor is <cap. delta><phi>'. It is
obvious that if the charge collected on the capacitor is Q, the battery
has transmitted it through itself under the potential difference
<cap. delta><phi> and then has given it an energy equal to
Q<cap. delta><phi>. But Eq.(3) states that the electrostatic potential
energy of the capacitor is 1/2 Q<cap. delta><phi>'. According to the
conservation law of energy then we must have Q<cap. delta><phi> =
1/2 Q<cap. delta><phi>' or <cap. delta><phi>' = 2<cap. delta><phi>.
A simple physical reasoning shows this fact too: When stating that the
electrostatic potential energy between the two conductors of the
capacitor is <cap. delta><phi>' we mean that supposing that all the
capacitor charges are fixed, if supposedly a one-coulomb external point
charge starts to move from one of the two conductors under the
influence of the electrostatic force of the capacitor until it reaches
the other conductor, the work performed on it by this force will be
<cap. delta><phi>', without any change in the charges on the conductors.
But if we suppose that the magnitude of the charge on each conductor
of the above capacitor is one coulomb and it is possible that charges
separate from a conductor and moving in the space between the two
conductors reach the other conductor, then the total work performed on
this one-coulomb charge by the electrostatic force of the capacitor
will not be certainly equal to <cap. delta><phi>', because with each
transmission of some part of the charge, magnitude of the charge on each
conductor (and consequently the electrostatic field between the two
conductors) is decreased and does not remain unchanged as before.
The above argument shows that this work will be 1/2 <cap. delta><phi>',
because this is in fact the same work done by the battery for charging
the capacitor being conserved in the capacitor in the form of potential
energy which is being released now. We show this matter in an analytical
manner too: Suppose that our capacitor is a parallel-plate one and its
charge is Q. If a separate Q-coulomb charge travels from a plte to the
other one, the work performed on it will be
Q d
QEd = Q ---------- d = ---------- Q{2}, (38)
<epsilon>A <epsilon>A
while for calculating the work performed on the charge of the capacitor
itself being plucked bit by bit traveling from a plate to the other one,
we should say that the work performed on a differential charge -dQ (note
that dQ is negative), similar to (38), is
Q+dQ d
(-dQ)Ed = -dQ ---------- d = - ---------- (Q+dQ)dQ.
<epsilon>A <epsilon>A
Sum of these differential works is
<integral from Q=Q to 0>-d/(<epsilon>A) (Q+dQ)dQ = 1/2 d/(<epsilon>A) Q{2}
which is half of the previous work (shown in Eq.(38)).
Thus we should expect to have 2<cap. delta><phi>=d/(<epsilon>[0]A)Q when
a battery with the potential difference <cap. delta><phi> has charged a
parallel-plate capacitor, while hitherto it is thought that
<cap. delta><phi>=d/(<epsilon[0]>A)Q. Since all the parameters of both
the recent relations are measurable (<cap. delta><phi> by voltmeter), then
the truth or untruth of each can be tested practically.
We should notice a point. When connecting a voltmeter to the two conductors
of a charged capacitor, it measures <cap. delta><phi> not <cap. delta><phi>',
because its operation is based on passing a weak electric current through a
circuit in the instrument and measuiring the potential difference between
the two ends of the circuit; and passing of a current means in fact the
same being plucked of the capacitor charge bit by bit from the conductors,
and then the voltmeter measures <cap. delta><phi>.
We should also say that there is no need that in the existent calculations
of electrical cicuits the potential difference of each capacitor to be
made double, because in these calculations the same <cap. delta><phi>
has been in fact intended not <cap. delta><phi>', because the electric
current passing through the circuit including the capacitor is the same
process of gradual loading and unloading of the capacitor, not passing of
charge through the space between the two conductors of the capacitor
retaining the capacitor charge unchanged. Therefore, it is proper to give
<cap. delta><phi> a name other than the electrostatic potential difference
which is the name of <cap. delta><phi>'. Let's call it (ie <cap.delta><phi>)
as circuital potential difference of the capacitor. In this manner when
it is necessary to apply closed circuit law we must consider just this
circuital potential difference when passing the capacitor not its
electrostatic potential difference.
Now, again, consider a closed circuit of a battery, with the potential
difference <cap. delta><phi>, and a capacitor, with the capacitance C.
Let's investigate the usual method of analysis of RC (or generally RLC)
circuits and see what the difficulty is in it. Without missing anything
we suppose that the circuit has no resistance (ie R=0). When a differential
electric charge dQ passes through the battery causes a differential change
in the electrostatic energy of the capacitor. In the first instance it
seems that when the differential charge dQ passes through the battery it
gains the differential energy <cap. delta><phi>dQ which, as a rule
according to the conservation law of energy, this same energy must be
conserved in the capacitor in the form of d(Q{2}/(2C)), and then
<cap. delta><phi>dQ=d(Q{2}/(2C)) ==> <cap. delta><phi>dQ=(Q/C)dQ ==>
<cap. delta><phi>=Q/C ==> <cap. delta><phi>-Q/C=0
which is just the same the same result which we could obtain from the
closed circuit law by traveling one time round the circuit if the
potential difference between the two conductors of the capacitor was
taken electrostatic potential difference, ie <cap. delta><phi>'=Q/C, not
circuital potential difference, ie Q/(2C)! The difficulty is that the
relation <cap. delta><phi>dQ=d(Q{2}/(2C)) is not necessarily true, for this
reason: If we had a mathematical relation, in the form of an equality,
between the energy given by the battery and the electrostatic energy stored
in the capacitor (ie Q{2}/(2C)), we could differentiate from each side
of the equality relation and understand that the change of energy in the
capacitor in the form of d(Q{2}/(2C))(=Q/CdQ) is exactly arising from what
the differential change in the battery. But since there is no such a
relation we cannot necessarily infer that change of energy in the
capacitor in the form of Q/CdQ is arising from passing of the charge dQ
through the battery and consequently from differential change of
<cap. delta><phi>dQ in the energy given by the battery, because eg by
writing Q/(2C)(2dQ) instead of Q/CdQ we can claim that this change of
energy in the capacitor is arising from passing of the charge 2dQ through
the battery and consequently from differential change of
<cap. delta><phi>(2dQ) in the energy given by the battery (ie <cap. del
ta><phi>(2dQ)=Q/(2C)(2dQ)), and the previous reasonings showed that
incidentally this is the case.
Thus, we should bear in mind that in the analysis of RLC circuits we
must attribute only the circuital potential difference, ie Q/(2C), not
the electrostatic potential difference, ie Q/C, to the capacitor of the
circuit. (Refer to the discussion of RLC circuit in this article.)
It is necessary to note the influence that inattention to the above-mentioned
problem (ie difference between <cap. delta><phi> and <cap. delta><phi>')
has on the results of the experiments of Millikan and Thomson for
determining charge and mass of the electron (and similarly positive ions).
In the experiment of Millikan the electric charge of each
charged oil droplet is proportional to k/E in which k is the coefficient
of proportion of Stokes and E is the electrostatic field between the two
plates of the parallel-plate capacitor used in the experiment.
As we know E between the two plates of a parallel-plate capacitor is
equal to the electrostatic potential difference <cap. delta><phi>' divided
by the distance d between the two plates. So the charge of each droplet
is proportional to k/<cap. delta><phi>'. But for practical determination
of <cap. delta><phi>' the potential difference read by the voltmeter
connected to the plates of the capacitor is considered erroneously,
while as we said this potential difference, <cap. delta><phi>, which we
called it as circuital potential difference, is half of <cap. delta><phi>'.
In other words as a rule the quantity so far recognized as the charge of
a droplet should be two times larger than the real charge of the droplet
and then the electron's charge obtained from the numerous repetitions
of the experiment of Millikan should be really half of what is at present
accepted as the charge of electron.
But this is not the case because the experiment of Millikan plainly lacks
sufficient accuracy (and a tolerance up to half of the real amount seems
natural for it because certainly it is unlikely that the electrons are
added or deducted only one by one). In fact it seems that the results of
this experiment have been adapted in some manner for being in conformity
with the results of the exact experiment of determination of electric
charge of electron by X-ray. (As we know in this experiment the wavelength
of X-ray can be determined by its diffraction via a diffraction grating
with quite known specifications, and then having this wavelength and Bragg's
equation and analyzing the diffraction of the ray via a crystal lattice
the lattice spacing, d, of the crystal can be determined; thereupon
considering the molecular mass and crystal density Avogadro's number N[0]
can be calculated with sufficient accuracy and using it in the formula
F=N[0]e, in which F is the Faraday constant and e is the charge of electron,
e can be obtained which is the same that has been accepted at present as
the charge of the electron.)
In the experiment of Thomson too, for evaluation of q/m related to the
charge and mass of the electron in the cathodic ray, this quantity, ie
q/m, is obtained proportional to the electrostatic field E between the two
plates of the parallel-plate capacitor through which the cathodic ray passes.
But again for practical determination of E the above-mentioned error is
repeated and while E is really equal to <cap. delta><phi>'/d the amount
read on the voltmeter, <cap. delta><phi>, (which is in fact equal to
1/2<cap. delta><phi>') is set instead of <cap. delta><phi>'. In other
words as a rule the quantity hitherto considered as q/m of the electron
(in the experiment of Thomson) should be half of its real amount. Then,
to obtain the real value of q/m we must multiply the value accepted
presently as q/m by 2.
But here we should say that it seems that this experiment (or any other
similar one) is not accurate in determining q/m of electron or positive
ions since in it a shooting motion has been assumed for the electron in
the cathodic ray (or for the positive ion in the positive ray), while
as explained in detail in the 12th article of this book we must consider
for it a longitudinal wave motion in the gas medium existent in the tube
without any charge transferring, and it seems that such a wave motion,
although has many similarities with the shooting motion, is not exactly
the same shooting motion and has difference with it. Thus, it is necessary
to doubt what has been accepted as the mass of electron.
Hamid V. Ansari
Biography of the author:
The author has passed his high school and university courses in Iran.
He has been interested in studying the physical subjects, which he has
deeply loved them, fundamentally, basically and with thoroughly conversance.
He has many investigations on different grounds of physics that the most
important ones (but not all) of them have been collected in this book.
He has been studying physics not as a necessity to obtain academic papers
but because of his strong love for it, and for this reason firstly he
has not been content with only what he had to study academically and
has entered any field he liked and had possibilities for investigation
in it, and secondly he has not been particular about Iranian university
degrees which according to him are not the signs of loving investigations
in physics.
Contrary to his strong liking for discovering, innovation and genuine and
fundamental investigations in physics, due to his poverty he has not been
able to find way to the active and searching world of the experimental and
research physics, and it is several years that his great talent on this
ground is being wasted following earning livelihood in Iran in some works
irrelevant to university investigations, educations and experimentations.
He requests scincerely of all individuals of the society of physics,
particularly those who can help him, that if after careful studying of
the articles of this book they reach the truth of his claim to have a
talent capable of flourishing widely arousing the motive for many
world-wide physical researches and experiments, not to hesitate to help
him to be placed in the flow of the world physics; for example by giving
him a scholarship to a proper university for a graduate course.
Particularly at present he is greatly interested in practical and
experimental investigations on the ground of nuclear physics and
discovering the facts hidden inside the matter.
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