15 Electrical Storage

CHAPTER 15

Electrical Storage

We now turn to a consideration of the storage of uncharged electrons (electric current), a subject that was not considered earlier because it was more convenient to wait until after the nature of electric charges was clarified.

The basic requirement for storage is a suitable container. Any conductor is, to some extent, a container. Let us consider an isolated conductor of unit cross section, a wire. This conductor has a length of n units, meaning that it extends through n units of extension space, the space represented in the reference system. Each of these units of the reference system is a location in which a unit of actual space (that is, the spatial component of a motion) may exist. In the absence of an externally applied electric voltage, the wire contains a certain concentration of uncharged electrons (actual units of space), the magnitude of which depends on the composition of the material of the conductor, as explained in Chapter 11. If this wire is connected to a source of current, and a very small voltage is applied, more uncharged electrons flow into the wire until all of the units of the spatial reference system that constitute the length of the wire are occupied. Unless the voltage is increased, the inward flow ceases at this point.

When the wire is fully occupied, the aggregate of electrons could be compared to an aggregate of atoms of matter in one of the condensed states. In these states all of the units of extension space within the limits of the aggregate are occupied, and no further spatial capacity is available. But if a pressure is applied, either an internal pressure, as defined in Chapter 4, or an external pressure, the inter-atomic motions are extended into time, and the addition of the spatial equivalent of this time allows more atoms to be introduced into the same section of the extension space represented in the reference system, increasing the density of the matter (the number of mass units per unit of volume of extension space) beyond the normal equilibrium value.

This ability of physical phenomena to extend into time when further extension into space is prevented is a general property of the universe that results from the reciprocal relation between space and time. The scope of its application is limited, however, to those situations in which a spatial response to an applied force is not possible. In the example just discussed, the compression of solid matter, the obstacle to further inward movement in space is the discrete unit limitation on subdivision. In a wide variety of astronomical phenomena that will be considered in Volume III, the obstacle is the limit on one-dimensional spatial speed. Here, in the electrical storage process, the obstacle is the fixed relation between the unit of actual space and the unit of extension space. An n-unit section of the extension space represented in the reference system can contain n units of actual space, and no more.

If a voltage is applied to force additional electrons into the fully occupied section of wire, the excess electrons are pushed out into time, where they occupy positions in the spatial equivalent of that time. This penetration into time can only be accomplished by application of a force, as the concentration of uncharged electrons in time is already at an equilibrium level. If the voltage is reduced or eliminated, the restoring force tending to bring the electron concentration back into equilibrium reverses the flow, and the excess electrons move back out of the wire. Application of a positive* voltage similarly withdraws electrons from the wire and from equivalent space.

As we have seen in the preceding pages of this and the earlier volume, the region of time beyond the unit of space is two-dimensional. The concentration of excess electrons and the effective voltage therefore decrease in direct proportion to the distance from the wire at a rate determined by basic physical factors and the dimensions of the wire (or other conductor), reaching the zero level at a specific distance.

Let us consider a case in which a conductor is subjected to a voltage differential of 2V, and the voltage in equivalent space surrounding each terminal reaches zero at a distance s from the terminal. As long as the terminals (electrodes) are separated by a distance greater than 2s, the electron storage, the quantity of current that can be withdrawn at the positive* terminal and introduced at the negative* terminal, is independent of the location of those terminals. However, if the separation is reduced to less than 2s, a portion of the volume of equivalent space from which the electrons are being withdrawn coincides with the volume of equivalent space into which electrons are being introduced. The excess and deficiency of electrons in this common volume cancel each other, decreasing the net excess or deficiency at the terminals, and thereby reducing the voltage. This means that where the separation of the terminals is reduced below 2s, the same amount of storage will take place at a lower voltage, or alternatively, a greater amount of storage will be possible at the same voltage.

The relations involved in the storage of current (uncharged electrons) are illustrated in Figure 21.

Figure 21

When the terminals are separated by the distance 2s, the full voltage drop, V, takes place at each terminal. The electron excess at the negative* terminal, which we will call E, is proportional to V. If the separation between the terminals is decreased to 2xs, there is an overlap of the equivalent volumes to which the excess and deficiency of electrons are distributed, as indicated above. The effective voltage then drops to xV. At this point, the electron concentration corresponding to xV is in the equivalent volume at the negative* terminal, while the balance of the total electron input represented by E is in the common equivalent volume, where the net concentration of excess electrons is zero. If the voltage is reduced, the electrons from the common equivalent volume, and from the volume related to the negative* terminal only, flow out of the system in the same proportions in which they entered. Thus the storage capacity at a separation 2xs and voltage xV is the same as that at a separation 2s and voltage V. Generalizing this result, we may say that the storage capacity, at a given voltage, of a combination of positive* and negative* electrodes in close proximity is inversely proportional to the distance between them.

The ability of a conducting wire to accept additional electrons when subjected to a voltage makes it available as a container in which uncharged electrons (units of electric current) can be stored and withdrawn as desired. Such storage has some uses in electrical practice, but it is inconvenient for general use. More efficient storage is made possible by a device that contains the necessary components in a more compact form. In this device, a capacitor, two plates, each with an area s2, are separated by a distance s’. Each plate is equivalent to s2 conductors of unit cross section. Thus the storage capacity of a capacitor at a given voltage is directly proportional to the plate area and inversely proportional to the distance between the plates. This storage capacity is called the capacitance, symbol C. Since it has the dimensions of space (s2/s’ = s), it can be calculated directly from the geometrical dimensions of the capacitor. The centimeter has been used as a unit, although the present practice is to use a special unit, the farad.

If a capacitor is connected to a current supply, the effective voltage, a force (t/s2), pushes the uncharged electrons that constitute the current into the capacitor until the concentration corresponding to that voltage is reached. The space-time dimensions of the product are t/s2 × s = t/s. This is inverse speed, or energy. It is not a charge, on the basis of the definition of charge given in this work, but since electric charge has the dimensions of energy, t/s, the quantity stored is equivalent to charge. To minimize the deviations from currently accepted terminology, we will call it a capacitor charge. The magnitude of the storage can be expressed by the equation Q = CV, where Q is the capacitor charge, C is the capacitance, and V is the voltage differential across the plates of the capacitor.

The unit of capacitance, the farad, is defined as one coulomb per volt. The volt is one joule per coulomb. These are units of the SI system, which will be used in most of the subsequent discussion of electricity and magnetism, rather than the cgs system of measurement that is in general use in these volumes, the reason being that a substantial amount of clarification of the physical relations in these areas has been accomplished in very recent years, and most of the current literature relating to these subjects utilizes the SI system.

Unfortunately, this recent clarification of the electrical and magnetic situations has not extended to some of the most fundamental issues, including the many problems introduced into electrical theory by the failure to recognize the existence of uncharged electrons and the consequent lack of distinction between electric quantity and electric charge. As we saw in Chapter 9, the unit of electric quantity is a unit of space (s). We find that the unit of electric charge is a unit of energy (t/s). In current practice, both of these quantities are expressed in the same measurement unit, esu (cgs system) or coulombs (SI system). Now that the electric charge has been introduced into our subject matter, we will have to make the distinction that current theory does not recognize, and instead of dealing only with coulombs, we will have to specify coulombs (s) or coulombs (t/s). In this work the symbol Q, which is currently being used for both quantities, will refer only to electric charge, or capacitor charge, measured in coulombs (t/s). Electric quantity, measured in coulombs (s) will be represented by the symbol q.

Returning now to the question as to the quantities entering into the capacitance, the volt, a unit of force, has the space-time dimensions t/s2. Since capacitance, as we have now seen, has the dimensions of space, s, the coulomb, as a product of volts and farads has the dimensions t/s2 × s = t/s. But the coulomb as the quotient of joules/volts, has the dimensions t/s × s2/t = s. Thus the coulomb that enters into the definition of the farad is not the same coulomb that enters into the definition of the volt. We will have to revise these definitions for our purposes, and say that the farad is one coulomb (t/s) per volt, while the volt is one joule per coulomb (s).

The confusion between quantity (s) and charge (t/s) prevails throughout the electrostatic phenomena. In most cases this does not result in any numerical errors, because the calculations deal only with electrons, each of which constitutes one unit of electric quantity, and is capable of taking one unit of charge. Thus the identification of the number of electrons as a number of units of charge instead of a number of units of quantity does not alter the numerical result. However, this substitution does place a roadblock in the way of an understanding of what is actually happening, and many of the relations set forth in the textbooks are incorrect.

For instance, the textbooks tell us that E = Q/s2. E, the electric field intensity, is force per unit distance, and has the space-time dimensions t/s2 × 1/s = t/s3. The dimensions of Q/s2 are t/s × 1/s2 = t/s3. This equation is therefore dimensionally correct. It tells us that, as we would expect, the magnitude of the field is determined by the magnitude of the charge. On the other hand, this same textbook gives the equation expressing the force exerted on a charge by the field as F = QE. The space-time dimensions of this equation are t/s2 = t/s × t/s3. The equation is therefore invalid. In order to arrive at a dimensional balance, the quantity designated as Q in this equation must have the dimensions of space, so that the equation in space-time form will become t/s2 = s × t/s3. in this case, then, the Q term is actually q (quantity) rather than Q (charge), and the applicable relation is F = qE.

The error due to the use of Q instead of q enters into many of the relations involving capacitance, and has introduced considerable confusion into the theory of these processes. Since we have identified the stored energy, or capacitor charge, as dimensionally equivalent to charge, Q, the capacitance equation in its customary form, Q = CV, reduces to t/s = s × t/s2, which is dimensionally consistent. The conventional form of the energy (or work, symbol W) equation is W = QV, reflecting the definition of the volt as one joule per coulomb. If CV is substituted for Q in this equation, as would appear to be justified by the relation Q = CV, the result is W = CV2. This equation is not dimensionally valid, but it and its derivatives can be found throughout the scientific literature. For instance, the development of theory in this area in one current textbook46 begins with the equation dW = VdQ for the potential energy of a charge, and by means of a series of substitutions of presumably equivalent quantities eventually arrives at an expression for energy in terms of E, the electric field intensity, and As, the volume occupied by the electric field. The first column of the accompanying tabulation shows the expressions that are equated to energy in the successive steps in this development. As indicated in the second column, the dimensional error in the first equation carries through the entire sequence, and the space-time dimensions of these expressions remain at t2/s3 instead of the correct t/s.

In Textbook Correct
QV t/s × t/s2 = t2/s3 qV s × t/s2 = t/s
Q/CdQ t/s × 1/s × t/s = t2/s3 Q/C dq t/s × 1/s × s = t/s
CV2 s × t2/s4 = t2/s3 qV s × t/s2 = t/s
E2 As t2/s6 × s2 × s = t2/s3 E(q/s2) As t/s3 × s/s2 × s2 × s = t/s

The error in this series of expressions was introduced at the start of the theoretical development by a faulty definition of voltage. As indicated earlier, the volt is defined as one joule per coulomb, but because of the lack of distinction between charge and quantity in current practice, it has been assumed that the coulomb entering into this definition is the coulomb of charge, symbol Q. In fact, as brought out in the previous discussion, the coulomb that enters into the energy equation is the coulomb of quantity, which we are denoting by the symbol q. The energy equation, then, is not W = QV, but W = qV.

The correct terms and dimensions corresponding the those in the first two columns of the tabulation are shown in columns 3 and 4. Here the term Q in the first two expressions, and the term CV which was substituted for Q in the last two have been replaced by the correct term q. As indicated in the tabulation, this brings all four expressions into agreement with the correct space-time dimensions, t/s, of energy. The purely numerical terms in all of these expressions were omitted from the tabulation, as they have no bearing on the dimensional situation.

When the full capacity of the capacitor at the existing voltage is reached, the opposing forces arrive at an equilibrium, and the flow of electrons into the capacitor ceases. Just what happens while the capacitor is filling or discharging is something that the theorists have found very difficult to explain. Maxwell found the concept of a “displacement current” essential for completing his mathematical treatment of magnetism, but he did not regard it as a real current. “This displacement does not amount to a current,” he says, ” but it is the commencement of a current.” He describes the displacement as “a kind of elastic yielding to the action of the force.”47 Present-day theorists find this explanation unacceptable because they have discarded the ether that was fashionable in Maxwell’s day, and consequently have nothing that can “yield” where the plates of a capacitor are separated by a vacuum. The present tendency is to regard the displacement as some kind of a modification of the electromagnetic field, but the nature of the hypothetical modification is vague, as might be expected in view of the lack of any clear understanding of the nature of the field itself. As one textbook puts it, “The displacement current is in some ways the most abstract concept mentioned in this book so far.”48 Another author states the case in these words:

If one defines current as a transport of charge, the term displacement current is certainly a misnomer when applied to a vacuum where no charges exist. If, however, current is defined in terms of the magnetic fields it produces, the expression is legitimate.49

The problem arises from the fact that while the physical observations and the mathematical analysis indicate that a current is flowing into the space between the plates of the capacitor when that space is a vacuum, as well as when it is occupied by a dielectric, such a current flow is not possible if the entities whose movement constitutes the current are charged electrons, as currently assumed. As stated in the foregoing quotation, there are no charges in a vacuum. This impasse between theory and observation that now prevails is another of the many items of evidence showing that the electric current is not a movement of charged particles.

Our analysis shows that the electrons do, in fact, flow into the spatial equivalent of the time interval between the plates of the capacitor, but that these electrons are not charged, and are unobservable in what is called a vacuum. Aside from being only transient, this displacement current is essentially equivalent to any other electric current.

The additional units of space (electrons) forced into the time (equivalent space) interval between the plates increase the total space content. This can be demonstrated experimentally if we introduce a dielectric liquid between the plates, as the increase in the amount of space decreases the internal pressure, the force per unit area due to the weight of the liquid. For this purpose we may consider a system in which two parallel plates are partially immersed in a tank of oil, and so arranged that the three sections into which the tank is divided by the plates are open to each other only at the bottom of the tank. If we now connect the plates to a battery with an effective voltage, the liquid level rises in the section between the plates. From the foregoing explanation it is evident that the voltage difference has reduced the pressure in the oil. The oil level has then risen to the point where the weight of the oil above the free surface balances the negative increment due to the voltage differential.

Because accepted theory requires the “displacement current” to behave like an electric current without being a current, conventional science has had great difficulty in ascertaining just what the displacement actually is. It is an essential element in Maxwell’s formulations, but some present-day authors regard it as superfluous. “All the physics of dielectrics could be discussed without ever bringing in the displacement vector,”50 says Arthur Kip. One of the principal factors contributing to this uncertainty as to its status is that the displacement is customarily defined and treated in electrostatic terms, whereas it is actually a manifestation of current electricity. In Maxwell’s equation for the displacement current, the current density, I/s2, and the time derivative of the displacement, dD/dt, are additive, and are therefore terms of an equivalent nature; that is, they have the same dimensions. The space-time dimensions of current density are s/t × 1/s2 = 1/st. The dimensions of D, the displacement, are then 1/st × t = 1/s. Its place in the capacitance picture is then evident. In the storage process, units of space, uncharged electrons, are forced into the surrounding equivalent space—that is, the spatial equivalent of time (t = 1/s)—and this inverse space, 1/s, becomes one of the significant quantities with which we must deal.

In the customary electrostatic treatment of the displacement, it is defined as D = ε0E, where E is the field intensity (an electrostatic concept) and ε0 is the permittivity of free space. Since the dimensions of E are t/s3, and we have now found those of D to be 1/s, the space-time dimensions of permittivity are 1/s × s3/t = s2/t. In current practice, however, the permittivity is expressed in farads per meter. This makes it dimensionless, since both the farad and the meter are units of space. We are thus faced with a conflict between the dimensional definition of permittivity expressed in the conventional unit and the definition derived from Maxwell’s relations, a definition that is consistent with the dimensions of displacement. The relation between the two in space-time terms, which is s2/t, shows where the difference originates, as this is the relation of the unit of electric current, s, to the electrostatic unit, t/s. The farad per meter is an electrostatic unit, while the s2/t dimensions for permittivity relate this quantity to the electric current system.

Permittivity is of importance mainly in connection with non-conducting substances, or dielectrics. If such a substance is inserted between the plates of a capacitor, the capacitance is increased. The rotational motions of all non-conductors contain motion with space displacement. It is the presence of these space components that blocks the translational motion of the uncharged electrons through the time components of the atomic structure, and makes the dielectric substance a non-conductor. Nevertheless, dielectrics, like all other ordinary matter, are predominantly time structures; that is, their net total displacement is in time. This time adds to the time of the reference system, and thus increases the capacitance.

From this explanation of the origin of the increase, it is evident that the magnitude of the increment will vary by reason of differences in the physical nature of the dielectrics, inasmuch as different substances contain different amounts of speed displacement in time, arranged in different geometrical patterns. The ratio of the capacitance with a given dielectric substance between the plates to the capacitance in a vacuum is the relative permittivity, or dielectric constant, of the substance.

The dielectric constants of most of the common dielectric substances—Class A dielectrics, as they are called—show little variation at low frequencies under ordinary conditions.51 This indicates that permittivity is an inherent property of the substance, a consequence of its composition and structure, rather than of its relation to the environment. This is consistent with the theoretical explanation given above.

Conventional theories of dielectric phenomena are based on the premise that these phenomena are electrostatic in nature. It should be understood, however, that all theories which depend on the existence of electric charges in electrically neutral materials cannot be other than hypothetical. Furthermore, conventional science has no comprehensive electrostatic theory of dielectrics. As expressed by W. J. Duffin:

It is important to realize that calculations of fields due to, and forces on, charge distribution are performed on a model and the results compared with experiment…different models are required to account for different sets of experimental results.52

In the model that is applied to the capacitance problem it is assumed (1) that positive* and negative* charges exist in the electrically neutral dielectric, (2) that “small movements of the charges have taken place in opposite directions,” and (3) that these movements produce the “polarization which we believe takes place (italics added).”53 As this statement by Duffin concedes, there is no direct evidence of the polarization that plays the principal role in the theory. The entire “model” is hypothetical.

Clarification of the dimensions of the quantity known as permittivity eliminates the static charges from the mathematics of the electrical storage process, and thereby cuts the ground out from under all of the electrostatic models. The customary mathematical treatment is carried out in terms of four quantities, the displacement, D, the polarization, P, the electric field intensity, E, and the permittivity, ε0. These quantities, the investigators tell us, are related by the expression P = Dε0E. We have already seen, earlier in this chapter, that the space-time dimensions of D are 1/s, and those of the permittivity, ε0,are s2/t. The dimensions of the quantity ε0E are then s2/t × t/s3= 1/s. It follows that the dimensions of P are also 1/s.

We thus find that all of the quantities entering into the dielectric processes are quantities related to the electric current: the electric quantity (s), the capacitance (s), the displacement (1/s), the polarization (1/s), and the quantity ε0E, which likewise has the dimensions 1/s. No quantity with the dimensions of charge (t/s) has any place in the mathematical treatment. The language is that of electrostatics, using terms such as “polarization,” “displacement,” etc., and an attempt has been made to introduce electrostatic quantities by way of the electric field intensity, E. But it has been necessary to couple E with the permittivity, ε0, and to use it in the form ε0E, which, as just pointed out, cancels the electrostatic dimension of E. Electric charges thus play no part in the mathematical treatment.

A similar attempt has been made to bring the electric field intensity, E, into relations that involve the current density. Here, again, the electrostatic quantity, E, is out of place, and has to be removed mathematically by coupling it with a quantity that converts it into something that has a meaning in electric current phenomena. The quantity that is utilized for this purpose is conductivity, symbol σ, space-time dimensions s2/t2. The combination σE has the dimensions s2/t2 × t/s3 = 1/st. These are the dimensions of current density. Like the expression ε0E, previously discussed, the expression σE has a physical meaning only as a whole. Thus it is indistinguishable from current density. The conventional model brings the field intensity into the theoretical picture, but here, again, it is necessary to remove it by a mathematical device before the theory can be applied in practice.

In those cases where the electric field intensity has been used in dealing with electric current phenomena, without introducing an offsetting quantity such as σ or ε0, the development of theory leads to wrong answers. For example, in their discussion of the “theoretical basis of Ohm’s law,” Bleaney and Bleaney say that “when an electric field strength E acts on a free particle of charge q, the particle is accelerated under the action of the force,” and this, “leads to a current increasing at the rate dJ/dt = n (q2/m) E,”54 where n is the number of particles per unit volume. The space-time dimensions of this equation are 1/st × 1/t = 1/s3 × s2 × s3/t3 × t/s3. Thus the equation is dimensionally balanced. But it is physically wrong. As the authors admit, the equation “is clearly at variance with the experimental observation.” Their conclusion is that there must be “other forces which prevent the current from increasing indefinitely.”

This fact that the key element of the orthodox theory of the electric current, the hypothesis as to the origin of the motion of the electrons, is “clearly at variance” with the observed facts is a devastating blow to the theory, and not all of its supporters are content to simply ignore the contradiction. Some attempt to find a way out of the dilemma, and produce explanations such as the following:

When a constant electric field E is applied each electron is accelerated during its free path by a force -e E, but at each collision it loses its extra energy. The motion of the electrons through the wire is thus a diffusion process and we can associate a mean drift velocity v with this motion.55

But collisions do not transform accelerated motion into steady flow. If they are elastic, as the collisions of the electrons presumably are, the acceleration in the direction of the voltage gradient is simply transferred to other electrons. If the force Eq actually existed, as present-day electrical theory contends, it would result in accelerating the average electron. The authors quoted in reference 54 evidently recognize this point, but they fall back on the prevailing confidence that something will intervene to save the “moving charge” theory of the electric current from its multiplicity of problems; there “must be other forces” that take care of the discrepancy. No one wants to face the fact that a direct contradiction of this kind invalidates the theory.

The truth is that this concept of an electrostatic force (Eq) applied to the electron mass is one of the fundamental errors introduced into electrical theory by the assumption that the electric current is a motion of electric charges. As the authors quoted above bring out in the derivation of their electric current equation, such a force would produce an accelerating rate of current flow, conflicting with the observations. In the universe of motion the moving electrons that constitute the electric current are uncharged and massless. The mass that is involved in the current flow is not a property of the electrons, which are merely rotating units of space; it is a property of the matter of the conductor. Instead of an electrostatic force, t/s2, applied to a mass, t3/s3, producing an acceleration (F/m = t/s2 × s3/t3 = s/t2), what actually exists is a mechanical force (voltage, t/s2) applied to a mass per unit time, a resistance, t2/s3, producing a steady flow, an electric current (V/R = t/s2 × s3/t2 = s/t).

Furthermore, it is observed that the conductors are electrically neutral even when a current is flowing. The explanation given for this in present-day electrical theory is that the negative* charges which are assumed to exist on the electrons are neutralized by equivalent positive* charges on the atomic nuclei. But if the hypothetical electrostatic charges are neutralized so that no net charge exists, there is no electrostatic force to produce the movement that constitutes the current. Thus, even on the basis of conventional physical theory, there is abundant evidence to show that the moving electrons do not carry charges. The identification of the electric current phenomena with the mechanical aspects of electricity that we derive from the theory of the universe of motion now provides a complete and consistent explanation of these phenomena without recourse to the hypothesis of moving charged electrons.

As noted in Chapter 13, charged electrons are subject to the same forces that apply to their uncharged counterparts, as well as to those specifically appertaining to the charges. It would therefore be theoretically possible to apply a voltage and store these charged electrons in capacitors in the same manner as the uncharged electrons (electric current). In practice, however, the storage of charged electrons is accomplished in a totally different manner. A widely used electrostatic device is the Van de Graaf generator. In this generator, charged electrons are produced and sprayed onto to a moving belt of insulating material. The belt carries them to a storage unit in the form of a large hollow metal sphere. The electrons pass from the belt into the sphere, gradually building up a potential that may reach a level as high as several million volts.

In our examination of electric current phenomena in the preceding chapters we found that the electrons which constitute the current move from the regions of higher voltage (greater concentration or higher speed of the electrons) to regions of lower voltage. In the Van de Graaf generator, electrons of very low electrostatic potential on the belt pass into a container in which the potential may be in the million volt range. Obviously, we are dealing with two different things, both having the dimensions of force, and both customarily measured in volts, but physically unlike in some important respects.

It should now be evident why the term “potential” was not used in the preceding pages in connection with capacitor storage, or other electric current phenomena. The property of the electric current that we are calling “voltage” is the mechanical force of the current, a force that acts in the same manner as the force responsible for the pressure exerted by a gas. Electrostatic potential, on the other hand, is the radial force of the charges, which decreases rapidly with the distance. The potential of a charged electron (in volts) is very large compared to the contribution of the translational motion of that electron to the voltage. It follows that even where the potential is in the million volt range, the electron concentration in the storage sphere, and the corresponding voltage, may be low. In that event, the small buildup of the voltage in the electrode at the end of the belt is enough to push the charged electrons into the storage sphere, regardless of the high electrostatic potential.

Many present-day investigators realize that they cannot account for electric currents by means of electrostatic forces alone. Duffin, for instance, tells us that “In order to produce a steady current there must be, for at least part of the circuit, non-electrostatic forces acting on the carriers of charge.”13 His recognition that these forces act on “the carriers of charge,” the electrons, rather than on the charges, is particularly significant, as this means that neither the forces nor the objects on which they act are electrostatic. Duffin identifies the non-electrostatic forces as being derived “from electromagnetic induction” or “non-homogeneities such as boundaries between dissimilar materials. or temperature gradients.”

Since the electric currents available to both the investigators and the general public are produced either by electromagnetic induction or by processes of the second non-electrostatic category mentioned by Duffin (batteries, etc.), the non-electrostatic forces that admittedly must exist are adequate to account for the current phenomena as a whole, and there is no need to introduce the hypothetical electrostatic charge and force. We have already seen that the charge does not enter into the mathematics of the current flow and storage processes. Now we find that it has no place in the qualitative explanation of current flow either.

Addition of these further items of physical and mathematical evidence to those discussed earlier now provides conclusive proof that the mathematical structure of theory dealing with the storage of electric current is not a representation of physical reality. This is not an isolated case. As pointed out in Chapter 13, the conditions under which scientific investigation is conducted have had the effect of directing the investigation into mathematical channels, and the results that have been attained are almost entirely mathematical. As expressed by Richard Feynman:

Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics.56

The development of this mathematical structure of theory is an outstanding achievement, and it has had very important—even spectacular—practical results. However, these successes have fostered a tendency to forget that mathematics is not physics. It is a useful, perhaps indispensable, tool for the physicist, but physical phenomena are subject to a multitude of limitations that do not apply to the mathematics that are utilized to represent these phenomena, and consequently are not recognized unless they are identified physically. The mathematical representation of space, for example, can be “curved,” or otherwise modified, but this does not, in any way, assure us that physical space can be so modified. That question can be settled only by means of a purely physical investigation such as the one reported in this work, which finds that such a modification of extension space is impossible.

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