Maxwell's concept of a field
Certain
aspects of Faraday's ideas concerning the notion of a field were given a mathematical
precision and rigor through the efforts of James Clerk Maxwell. More specifically, the
field equations which were developed in Maxwell's work Treatise on Electricity and
Magnetism provided a means of describing the rate at which electrical and magnetic
fields changed in relation to time as well as the three axes of space. The character of
the relationship of the rates of change of the electrical and magnetic fields along the
temporal and spatial dimensions were captured by Maxwell in the form of partial
differential equations.
Partial
differential equations were more complex and sophisticated versions of the ordinary
differential equations that were employed by Newton to describe the rates of change of
distance between objects. The latter kind of equations revolved around just one
independent variable - namely, rates of change with respect, say, to the distance between
objects when calculating gravitational effects.
However,
when one was trying to determine the rates of change of electrical and magnetic fields,
one had to treat each of the spatial coordinates separately from one another. This is
required because the orientation or direction of a magnetic field is different from the
orientation or direction of the electrical current which is the source of that magnetic
field.
In addition,
the partial differential equations devised by Maxwell had to take into consideration the
fact that a magnetic field was actually equivalent to a moving electrical field, just as
an electrical field was equivalent to a moving magnetic field. This meant his equations
needed to keep track of the rate at which the position of a body changed in relation to
time since this reflected the property of motion which played such an important role in
linking magnetic and electrical phenomena.
Maxwell's
equations gave expression to certain aspects of the idea of a field which had been
introduced by Faraday. More specifically, Maxwell's equations described fields that were
manifestations of continuously distributed densities of electrical or magnetic charge
which constitute the field of force that is generated by a given material source. The way
in which variations in charge density manifest themselves as a continuous function of
changes in time and position within a given field is dependent on the character of the
source material with which the field is associated.
Maxwell's
concept of a field differed from Faraday's idea of a field of force. Although Maxwell
based his mathematical formulations on the experimental findings of Faraday, Maxwell did
not adopt the theoretical framework in which Faraday had embedded his empirical
observations. Whereas Faraday had maintained matter is but a manifestation of a more
fundamental field of force and, therefore, believed there was an equivalency between
matter and the field with which it was associated, Maxwell treated matter and its
associated field as separate entities.
Maxwell did
retain something of Faraday's field concept in as much as he believed bodies did not act
directly on other bodies. Thus, he agreed with Faraday that the action of bodies on other
bodies was mediated through a field.
On the other
hand, Maxwell kept something of a Newtonian flavor in his thinking by treating force as
something which was to be distinguished from matter. Moreover, like Newton, he didn't
conceive of, say, an electrical field as being a field of force as Faraday had done.
Maxwell thought in terms of mathematical formulations which allowed one to calculate the
quantity of force per unit charge which was present in a given field.
Yet, Maxwell
differed from both Newton and Faraday. He believed a field - in this case an electrical
one - occupies, in some unspecified sense, the same space as does the charge on which the
field acts, despite believing that matter and the field were two different things.
Maxwell
chose to reinterpret issues surrounding the field concept in terms of the idea of an
ether. More specifically, he believed the electromagnetic field could be construed in
terms of an ether which conformed to the laws of Newtonian mechanics.
Consequently, Maxwell assumed the ether possessed, among other things, mechanical
properties such as mass and elasticity.
Because the
ether was assumed to have a variety of mechanical properties, Maxwell believed phenomena
such as electromagnetism would be propagated through the ether field at a finite velocity.
Thus, he agreed with Faraday about the finite character of the propagation in the
electromagnetic field, but he differed with Faraday concerning the reasons why this was
the case.
Maxwell's
theoretical task became a matter of constructing a model of electromagnetic phenomena in
terms of the idea of a mechanical ether. He had to show how the mechanical ether was able
to induce phenomena to not only propagate at a finite velocity, but to propagate in a way
that was in agreement with what was already known about electrical and magnetic phenomena.
One of the
primary reasons for Maxwell's shift in perspective away from Faraday's field concept is
because the idea of an ether lent itself to exploitation by means of mechanical analogs in
a way that Faraday's fields of force did not. In other words, the idea of the
ether had heuristic value in as much as its somewhat amorphous character allowed Maxwell
to build a variety of mechanical properties into it that collectively were capable of
reflecting many of the structural characteristics of electrical and magnetic phenomena.
Indeed, Maxwell's mechanical model of the ether permitted him to provide a unified means
of describing phenomena such as electrical currents, static electricity, magnetism and the
inductive effects resulting from the interaction of electrical and magnetic properties.
The method of analogies
The method
of analogies played a fundamental role in helping Maxwell to develop his theory of the
electromagnetic field. Maxwell, of course, realized the mechanical analogs he was using to
represent electrical and magnetic phenomena were not correct. That is, he did not feel the
electromagnetic field possessed the sort of mechanical structures in which his model was
rooted. At the same time, he believed models which were false in certain ways still could
help uncover various aspects of the truth.
Thus, even
if, ultimately, his mechanical model did not accurately reflect the structural character
of the electromagnetic field as far as how certain effects and properties actually
(i.e., ontologically) were generated in that field, nonetheless, the model allowed him to
develop a clear grasp of the sorts of properties and relationships which existed in the
field. As a result, he would be in a position to transform his understanding into a
mathematical description which would accurately reflect the properties and relationships
that characterized the electromagnetic field.
When all is
said and done, Maxwell derived a set of equations that conformed to the laws of Newtonian
mechanics and which also permitted one to accurately reflect many aspects of the
structural character of electromagnetic fields. However, his model left open, if not
entirely unanswered, the identity of the ontological "mechanism" which was
responsible for the phenomena of electromagnetic fields.
Among the
mechanical analogs that Maxwell developed was the idea of vortices or eddies in the ether
medium. These gave representation to magnetic phenomena.
Vortices
were conceived of as being like flexible bars with a rough surface. He theorized that in
any given locality of the field such vortices revolved in the same direction about axes
which were roughly parallel with one another. However, as one moved from one locality in
the field to other localities of the field, the properties of the vortices could change
such that the rotational velocity of the vortex, or the direction of movement of the
vortex about its axis, could vary due to differences in conditions prevailing in different
localities of the field.
In addition,
Maxwell proposed that the vorticies were separated from one another by a medium made up of
small electrical balls. These balls revolved about their own axes in a direction opposite
to the direction of rotation of the vortices.
Movement of
these balls constituted an electrical current. Moreover, by linking the mechanics of
vortex behavior with the mechanics of the behavior of the electrical balls, Maxwell was
able to bring electrical and magnetic phenomena in contact with one another within the
context of the field.
The concept
of a displacement current assumed a role of central importance to Maxwell's
theory of the electromagnetic field. According to Maxwell, when a displacement current
occurred at a particular point in the field, this was due to a change in position of a
given electrical charge.
Such a
transition in electrical displacement would have, associated with it, a magnetic field
which would spread out at right angles from the path of the displacement current. Since
the magnetic field would be mirroring changes in the displacement current, the changes in
the magnetic field would bring about further electrical displacements as a result of the
moving magnetic field's capacity to generate electromotive forces of induction.
This
dialectic of electrical displacements, leading, in turn, to the emergence of a magnetic
field, which would lead to further electrical displacements, and so on, would go on for an
indefinite period of time and would spread throughout the field. However, neither Maxwell,
nor anyone since him, has been able to provide a tenable account of why displacement
currents should be able to generate the ensuing complex of interacting fields and waves
Some field
components and their hermeneutical counterparts
The
following terms form the basis for many of the components of Maxwell's equations:
(1)
‘j’ stands for the intensity of current at a given point in the field, and, in
Maxwell's mechanical model, it is calculated by the number of balls which pass a given
point per unit of time;
(2)
‘H’ represents the intensity of the magnetic force, which Maxwell measured by
calculating the speed of the vortex (the mechanical analog-image which Maxwell devised to
help him think about how magnetic force might manifest itself) at its surface. Maxwell
contended that magnetic force arises from the centrifugal force of the vortices:
(3)
‘mu’- gives expression to the magnetic permeability of a given field as measured
by the average density of the vortices in that field;
(4) (mu)H2
- the kinetic energy of the rotating vortices is proportional to this; the energy of
the magnetic field is expressed in terms of such kinetic energy;
(5)
‘E’ constitutes the part of the electromotive force which is due to induction.
In terms of Maxwell's mechanical model, when two adjacent vortices have differences in
rotational velocity, there will be a tangential force exerted on whatever electrical
particles exist between the vortices. This tangential force is E. Thus, for Maxwell, electromotive
force is a function of the field stresses which impinge on the mechanism (whatever its
identity turns out to be) which links together various kinds of motion in the field:
(6)
‘A’ refers to the vector potential or electrotonic state which is associated
with the momentum of the vortices in a field. Electromotive force is functionally
dependent on changes in the momentum of the vortices. Moreover, the electromagnetic
induction of currents emerges in contexts In which the changing velocities of various
vortices brings into play forces capable of generating such induction activity;
(7)
‘D’ is the label given to the displacement or elastic distortion to which any
given electrical ball is subjected. When electrical balls are rooted in a dielectric, the
mechanical model does not permit them to move, but the balls can be distorted by the
forces which are acting on them. How much distortion will occur will be a function of both
the character of the forces acting on the ball, as well as the elastic properties of the
ball. Maxwell demonstrated that a change in displacement is capable of generating a
magnetic field, just as a conduction current is capable of doing so. The displacement
current constitutes the rate at which the displacement component changes over time -
delta-D/delta-t
. In short, Maxwell held that electric displacement was functionally dependent on the
degree to which the mechanism connecting different modes of motion in the field displayed
elasticity in the context of forces which were impinging on that connecting mechanism;
(8)
‘Epsilon’ - is the dielectric constant or inductive capacity of a medium;
(9)
‘Psi’ - concerns the electrical potential or tension which is exerted by the
electrical particles on one another. This pressure is the source of charge in Maxwell's
model. The differential pressure which is exerted on the sides of an electrical particle
generates the part of the electromotive force which is due to static electricity. For
Maxwell, a charged body is the result of a net pressure being exerted on the surrounding
dielectric by the electrical particles of that body;
(10)
‘curl’ is the modern term (i.e., although the term is not from Maxwell,
it does refer to a concept of his) that is used to express the idea of a rotating torque
which is exerted on any small electrical ball which might be placed at a given point in a
force field. According to Maxwell, one can state the motion of the electrical balls (i.e.,
the electrical current) as a function of the differentials in vortex speed (known as
magnetic intensity) which occur from place to place in the field. The term 'curl' refers
to this function.
(11)
‘div’ refers to the divergence of a given current. According to Maxwell, the
divergence of the total current of a field is zero, and in this sense, divergence is a
mathematical property of the total current of any given field. In the context of field
exhibiting displacement currents, Maxwell interpreted the fact that the total current of a
field has zero divergence to mean that all of the total currents of the field form
complete or closed circuits.
(12)
‘rho’ - refers to the charge density of a conductor;
(13) ‘r’
represents the resistance in a conductor;
(14) ‘v’
is the velocity of a moving conductor;
j, E, D, H,
A and v are all vector quantities, whereas: ’psi’‘mu’,
‘rho’, and ‘r’ are all scalar quantities.
The sections
(a) - (j) on the next several pages constitute a few of the possible
hermeneutical/phenomenological counterparts to some of the foregoing concepts of Maxwell's
electromagnetic field theory. Moreover, like the latter theory, the ensuing concepts can
be incorporated into a set of field equations (These equations could not be translated
into HTML coding, so they do not appear here). However, the hermeneutical
field equations, unlike their electromagnetic counterparts, are, at the present time,
still qualitative. In order to become quantitative, some appropriate means of
operationalization would need to be discovered .
(a) The
dialectic of focus and horizon (whether in the context of point-structures, neighborhoods,
or latticeworks) is given expression through the exchange of phase quanta and the
establishment of phase relationship states. This dialectical involves cyclical/oscillating
activity which is manifested through different waveforms that are described by a set of
constraints and degrees of freedom.
The
displacement current is a waveform that alters the ratio of constraints and degrees of
freedom at a given point in the dialectic of focus and horizon. By altering the ratio at a
given point in the dialectical process, this sets in motion a chain of alterations of
ratios in various other point-structures, neighborhoods or latticeworks which are contiguous
to (that is, interact with) the ratio or set of ratios which have been altered by the
displacement current that gives expression to the effect of a given order-wave.
The effect
of the initial displacement current radiates outward from the path of the current, cutting
across other lines of force or phase relationships in either focus and/or horizon. This
results in switches of focus, as well as in the introduction of a variety of horizonal
(e.g., memory ) components. The introduction of horizonal material can lead to further
switches in focus, just as switches in focus can lead to the further introduction of
horizonal vectors, whether in the form of memories, or learning or emotion or beliefs or
values or motivations or desires or needs or problems.
(b)
Phenomenological waveforms and hermeneutical waveforms are generated by the dialectic of
focus and horizon. In the case of phenomenological waveforms, the focal/horizon dialectic
gives expression to a framework of reflexive awareness. Awareness is the focus, and the
tacit intuiting of the constraints of the range of such awareness forms the horizonal
boundaries. In the case of hermeneutical waveforms, the focal/horizonal dialectic is
expressed in terms of the activity of the hermeneutical operator (combining the operations
of: (1) identifying reference, (2) reflexive awareness, (3) characterization, (4) the
interrogative imperative, (5) inferential mapping, and (6) congruence functions) engage
the various horizonal considerations and work toward a hermeneutical orientation as this
operator cuts across the lines of force existing in both focus and horizon.
The
hermeneutical framework which exists at a given time gives equal expression to both focal
and horizonal components. The intensity, immediacy and resolution of focus is
counterbalanced by the depth, breadth and pervasiveness of horizon.
(c) The
hermeneutical/phenomenological counterpart to the idea of magnetic permeability might be
construed in terms of the receptivity or sensitivity of a given point-structure,
neighborhood, or latticework which is to be induced into dialectic activity by other
point-structures, neighborhoods, and latticeworks. There may be differences of active and
passive potential which are intrinsic features of a given point-structure, neighborhood or
latticework, and such features will affect the direction of dialectical currents which
arise.
In this
sense, some mediums are more permeable to forces of induction than are other mediums.
Moreover, different phenomenological/hermeneutical mediums will be preferentially oriented
toward (and, therefore, selectively permeable to) different forces of
phenomenological/hermeneutical induction.
(d) The
hermeneutical counterpart to the notion of charge density concerns the quality of
intensity which is associated with a given aspect of phenomenology or hermeneutical
activity.
(e) The
component of elasticity can be rendered in terms of a ratio of constraints to degrees of
freedom in the sense that the more degrees of freedom which are present, then, the more
elastic or flexible is the point-structure or neighborhood or latticework which displays
such a ratio. On the other hand, if a given structure manifests a preponderance of
constraints over degrees of freedom, then, it tends to be more rigid, or less elastic and
flexible.
Previously,
entropy was construed in terms of the ratio of constraints and degrees of freedom. One
could conceive of elasticity as a particular facet of entropy. Thus, hermeneutical
structures exhibiting a high degree of entropy will manifest a low degree of elasticity or
flexibility with respect to their capacity to exhibit congruence properties. Similarly,
hermeneutical structures exhibiting a low degree of entropy will display a high degree of
elasticity in the way their manifolds 'cover or reflect an issue.
(f) The
hermeneutical counterpart to the notion of curl may have to do with the set of tensors
which impinge on focus and generate an orientation through which focus is expressed. The
idea of orientation has something of the quality of torque about it.
Orientation
gives expression to a sort of hermeneutical twisting. This leads to cyclical activity in
and about an attractor basin whose shape is a function of the character of the tensor
forces which are present and impinging on focus. Furthermore, treating orientation as an
expression of a sort of torque force also suggests something of the difficulty which is
involved in bringing about changes in orientation because of the complexity and strength
of the tensor set which is generating such an orientation.
(g) The
hermeneutical counterpart for the property of resistance involves the aspects of a
spectrum of ratios of constraints and degrees of freedom that actively or passively resist
being induced into manifesting themselves in a given way under certain circumstances. As
such, resistance would seem to be a property of the particular qualities of a given
dialectic, rather than some autonomous, independent property which manifests itself in the
same way under all circumstances.
(h) The
ideas of divergence and free electricity have a hermeneutical counterpart in the ideas of bound
and unbound hermeneutical operators. Bound operators function according to the
constraints and degrees of freedom which color, shape, orient, direct, and organize the
operator's activity as a result of certain values, beliefs, ideas, commitments,
assumptions, emotions, and so on.
In this
sense, the bound hermeneutical operator represents something akin to the idea of the
property of divergence which indicates that the currents of a field are closed circuits.
The bound hermeneutical operator also constitutes something of a closed circuit since its
character is described by the set of constraints and degrees of freedom to which the
hermeneutical operator is bound through a given value, belief, attitude, idea, emotion,
and so on.
Unbound
hermeneutical operators, on the other hand, engage various aspects of the phenomenology of
the experiential field in a way that is, relative to bound operator activity, sensitive,
if not receptive, to the currents, eddies, vectors and other properties of that
phenomenology. However, unbound hermeneutical operators do not impose value structures
onto the experiential field as bound hermeneutical operators do. In this sense, unbound
hermeneutical operators are a bit like the idea of free electricity and have a certain
amount of latitude to be able to generate new circuits whose shape and character will be a
function of the dialectic between such unbound hermeneutical operators and whatever aspect
of phenomenology or ontology is being engaged.
(i) A
dielectric is a nonconductor of direct electrical current. It refers to the aspects of
resistance in a given medium to the conducting of an electrical current. Presumably, there
are elements in the phenomenology of the experiential field- or, more particularly, in the
hermeneutics of the phenomenology of the experiential field - which are resistant to the
conducting or induction or propagation of a hermeneutical current.
The question
is whether this resistance is a constant in different hermeneutical and phenomenological
contexts, and if this is the case, whether one could identify what it is in such mediums
which is contributing to the resistance of hermeneutical currents. In general terms, it
seems intuitively attractive to suppose there are elements in the intellectual, emotional,
sensory, and spiritual contexts which would be resistant to the conducting of certain
kinds of hermeneutical currents.
Biases,
prejudices, attitudes and beliefs are obvious examples in the context of intellectual
issues. The absolute refractory period of neuronal functioning is an example on the level
of biological activity. In addition, negative emotional forces such as envy, pride, anger,
and jealousy might serve as examples on the level of emotion which would pose an inherent
resistance to the propagating of hermeneutical currents involving, say, love, compassion,
generosity, humility, and so on.
(j)
Reflexive awareness, identifying reference, characterization, the interrogative
imperative, inferential mapping, congruence functions and emotions are all vector
quantities. Experiential intensity may be - depending on circumstances - either a scalar
or vector quantity.
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