Clarification of the structure of the gravitational equation and application of the new information to formulation of the primary force equation opens the door to an understanding of the Coulomb equation, F = QQ’/d^{2}, that expresses the electrostatic force. This equation is set up on an equivalent basis without a numerical coefficient; that is, the numerical value of the charge Q is defined by the equation itself. It would seem, therefore, that when the other quantities in the equation, force, F, and distance, d, are expressed in terms of the cgs equivalents of the natural units, Q should likewise take the cgs value of the appropriate natural unit. But the dimensions of charge are t/s, and the natural unit of t/s in cgs units is 3.334×10^{-11} sec/cm, whereas the experimental unit of charge has the different numerical value 4.803×10^{-10}. In conventional physics there is no problem here, as the unit of charge is regarded as an independent quantity. But in the context of the theory of the universe of motion, where all physical quantities are expressed in terms of space and time only, it has been a puzzle that we have only recently been able to solve. One of the new items of information that was derived from the most recent analysis of the gravitational equation, and incorporated into the primary force equation, is that the individual force equations deal with only one force (motion). The force (apparently) exerted by charge A on charge B and the force (apparently) exerted by charge B on charge A are not separate entities, as they appear to be; they are merely different aspects of the same force. The reasons for this conclusion were explained in the gravitational discussion.

A second point, also derived from the gravitational study reported in Chapter 14, although it could have been arrived at independently, is that there is a missing term in the usual statement of * each* of the force equations. This term, identified as 1/s × (s/t)^{n-1} in the primary force equation, must be supplied in order to balance the equation. In the gravitational equation it is an acceleration term. In the Coulomb equation it is reciprocal space, 1/s.

Here we encounter a difference between the two equations that we have been examining. In the gravitational equation the unit of mass is defined independently of the equation. In the Coulomb equation, however, the unit of charge is defined by the equation. Consequently, any term that is omitted from the statement of the equation is automatically combined with the charge, instead of having to be introduced separately, as was necessary in the case of the acceleration term of the gravitational equation. The quantity 1/s, which, as we have just seen, is required for a dimensional balance, therefore becomes a component of the quantity that is called “charge” in the statement of the equation. That quantity is actually t/s (the true dimensions of charge) multiplied by 1/s (the omitted term), which produces t/s^{2}.

The same considerations apply to the size of the unit of this quantity. Since the charge is not defined independently of the equation, the fact that there is only one force involved means that the expression QQ’ is actually Q^{½}Q’^{½}. It follows that, unless some structural factor (as previously defined) enters into the Coulomb relation, the value of the natural unit of Q derived from that relation should be the second power of the natural unit of t/s^{2}. In carrying out the calculation we find that a factor of 3 does enter into the equation. This probably has the same origin as the factors of the same size that apply to a number of the basic equations examined in Volume I. It no doubt has a dimensional significance, although a full explanation is not yet available.

The natural unit of t/s^{2}, as determined in Volume I, is 7.316889×10^{-6} sec/cm^{2}. On the basis of the findings outlined in the foregoing paragraphs, the value of the natural unit of charge is:

- Q = (3 × 7.316889×10

There is a small difference (a factor of 1.0032) between this value and that previously calculated from the Faraday constant. Like the similar deviation between the values for the gravitational constant, this difference in the values of the unit of charge is within the range of the secondary mass effects, and will probably be accounted for when a systematic study of the secondary mass relations is undertaken.

The equivalence of the scalar motions AB and BA, which plays an important part in the force relations, is also responsible for the existence of a unique feature of static electricity, the * induction* of charges. One of the basic characteristics of scalar motion, resulting from this equivalence is that it is indifferent to *location *in the reference system. From the vectorial standpoint, locations are very significant. A vectorial motion originating at location A and proceeding in the direction AB is specifically defined in the reference system, and is sharply distinguished from a similar motion originating at location B and proceeding in the direction BA. But since a scalar motion has magnitude only, a scalar motion of atom A toward atom B is simply a decrease in the distance between A and B. As such, it cannot be distinguished from a similar motion of B toward A. Both of these motions have the same magnitude, and neither has any other property.

Of course, the scalar motion *plus *the coupling to the reference system does have a specific location in that system: a specific reference point and a specific direction. But the coupling is independent of the motion. The factors that determine its nature are not necessarily constant, hence the motion AB does not necessarily continue on the AB basis. A change in the coupling may convert it to BA, or it may alternate between the two.

The rotational component of the scalar motion of a charged atom always maintains the same relation to an atom at another location. Half of the elements of that rotational motion are approaching the second atom, while the other half are receding in equivalent directions and at equivalent speeds. But this is not true of the rotational vibration that constitutes a charge. In this case the relation of the motion (charge) to the distant atom is continually changing; that is, the relative motion of the two atoms has the same vibratory character as the charge itself. As has been stated, a scalar motion A (such as a charge) toward or away from atom B is indistinguishable from a similar motion of B toward or away from A. The representation of this motion in the spatial reference system can therefore take either form.

Ordinarily, some redistribution of energy is required for a change from one representation of a motion to another, and such changes therefore do not usually take place in the absence of external forces. In fact, Newton’s first law of motion requires a motion in the direction AB to continue in that direction indefinitely unless acted upon by some force. However, there is an exception to this general rule because of the existence of a class of phenomena that we may call* zero energy processes. *Most of the physical processes that have been examined in the preceding pages either operate by application of energy, or occur spontaneously with release of energy. For instance, there is a force of cohesion between the atoms of a solid, and energy must be applied to separate them. If they are allowed to recombine, a corresponding amount of energy is liberated. But the various components of a combination of basic motions are not bound to each other in this manner in all cases. Often they are merely associated, and are free to separate or combine without gaining or losing energy.

One such zero energy process is the simultaneous creation or destruction of charges of the same magnitude and opposite polarity. It is the existence of this process, together with the equivalence of scalar motions AB and BA, that makes the induction of electric charges possible. As we saw earlier, all material objects contain a concentration of uncharged electrons, which are essentially rotating units of space. In each case where an electron exists in an atom of matter, the atom likewise exists in the unit of space that constitutes the electron. This might be compared to a solution of alcohol in water. The atoms of alcohol exist in the water, but it is equally true that the atoms of water exist in the alcohol.

Let us now consider an example in which a positively* charged body X is brought into the vicinity of an otherwise isolated metallic object Y. The scalar direction of the vibratory motion (charge) of atom A in object X is periodically reversing, and at each reversal the reference point of the motion of A relative to any atom B that is * free to move *is redetermined by chance; that is, the motion may appear in the reference system either as a motion of A toward B or a motion of B toward A. By means of this chance process, the motion is eventually divided equally between AB and BA.

An atom C that is located in extension space is *not *free to move because energy would be required for the motion. But atom B in object Y, which is located in electron space, is not subject to this energy restriction, as the rotational motions of the atom and the associated electron are oppositely directed, and the same motion that constitutes a positive* charge on the atom constitutes a negative* charge on the electron, because in this case it is related to a different reference point. The production of these oppositely directed charges is a zero energy process. It follows that atom B is free to respond to the periodic changes in the direction of the scalar motion originating at A. In other words, the positive* charge on atom A in object X *induces *a similar positive* charge on atom B in object Y and a negative* charge on the associated electron.

The electron is easily separable from the atom, and it is therefore pulled to the near side of object Y by the positive* charge on object X, leaving atom B in a unit of extension space, and with a positive* charge. The positions of the positively* charged atoms are fixed by the inter-atomic forces, and these atoms are not able to move under the influence of the repulsive forces exerted by charged object X, but the positive* charges are transferred to the far side of object Y by the induction process. The residual positive* charge on atom B induces a similar charge on a nearby atom D that is located in electron space. The electron at D, now with a negative* charge, is drawn to atom B, where it neutralizes the positive* charge and restores the atom to the neutral status. This process is repeated, moving the positive* charge farther from object X in each step, until the far side of object Y is reached.

Where the original charge on object X is negative*, a negative* charge is induced on the electron associated with atom B. This is equivalent to a positive* charge on the atom. In this case, the negatively* charged electron is repelled by the negative* charge on object X and migrates to the far side of object Y. The residual positive* charge on the atom is then transferred to the near side of this object by the induction process.

If metallic object Y is replaced by a dielectric, the situation is changed, because in this case the electrons no longer have the capability of free movement. The induced charge on the atom and the opposite charge on the electron (or vice versa) remain joined. It is possible, however, for this atom to participate in a relative orientation of motions with a neutral atom-electron unit with which it is in contact, the result being a two-atom combination in which the negative* pole of one atom is neutralized by contact with the positive* pole of the other, leaving one atom-electron unit positively* charged and the other negatively* charged (that is, the charge is on the electron).

The optimum separation between the unlike charges, when under the influence of an external charge, the condition that is reached when the carriers of the negative* charges are free to move, is the maximum. The situation in the two-atom combination is therefore more favorable than that in the single atom, and the combination takes precedence. A still greater separation is achieved if one or more neutral atoms are interposed between the atoms of the charged combination. Each event of this kind moves either the positive* or the negative* charge in the direction determined by the inducing charge. Thus the effect of an inducing charge on a dielectric is a separation of the positive* and negative* charges similar to, but less complete than, that which takes place in a conductor, because the length of the chains of atoms is limited by thermal forces.

On the basis of the foregoing explanation, the charges are *produced *by induction. The subsequent separation is accomplished by action of the inducing charge on the newly produced induced charges. Conventional theory of dielectrics is based on the concept of the nuclear atom, a hypothetical structure in which the components are held together by the attraction between positive* and negative* charges. It is assumed that these charges have a limited amount of freedom of movement, and can separate slightly on being subjected to the effect of an external charge. One observation that has been interpreted as supporting the assumption that pairs of positive* and negative* charges are always present in the atoms is that if a charged dielectric is subdivided, each of the parts contains both positive* and negative* charges. This is quite different from the behavior of charges in conductors. If a metallic object is cut perpendicular to the line of force while under the influence of an inducing charge, the two parts are oppositely charged, and will remain so after the inducing charge is removed. But if the same procedure is followed with a dielectric, both parts have positive* and negative* charges on the opposite sides, just as in the original object before separation. And when the inducing charge is removed, both parts revert to the neutral status. The current interpretation of these results, as expressed in a contemporary textbook, is this:

The inference to be drawn is that insulators contain charges which can move small distances so that attraction still occurs, but that they are bound in equal and opposite amounts so that no splitting of the body can separate two kinds of charge.

^{57}

The amount of separation of charges that could take place in the manner assumed by this theory is admittedly very small, and it is difficult to account for the generation of any substantial attractive or repulsive forces by this means. But forces of this nature actually do exist. Small static charges, usually produced by friction, are common in the terrestrial environment, and they produce effects that are quite noticeable. Merely walking across a carpeted room in cold, dry weather can build up enough charge to give one an uncomfortable sensation when he touches a metallic object and discharge occurs. Likewise, the behavior of the modern synthetic fabrics shows the effect of static charges, including the inductive effect, in a conspicuous, and often annoying, way. These fabrics behave in much the same way as charged conductors. They attract such things as bits of paper and chips of wood, and are themselves attracted by the furniture and walls of a room.

The discrepancy between the very small theoretical separation of charges and the relatively large inductive effect has forced the theorists to call upon collateral factors, such as the presence of contaminants, to explain the observations. For example, the following statement taken from a physics textbook refers to the ability of electrically charged non-conducting objects to pick up bits of paper and wood:

A chip of perfect insulator would show hardly any effect, but bits of wood and paper always have enough moisture to make them slightly conducting.

^{58}

The much greater separation of charges that results from the inductive process described in this chapter resolves this problem, while it remains consistent with the appearance of charges at both ends of each piece when a dielectric under the influence of an inducing force is separated. Before the separation takes place a substantial number of atoms of the dielectric exist in multi-atom combinations with positively* and negatively* charged ends. Although the separation of the charges in many of these combinations is large compared to the distance between atoms, it is very small compared to the dimensions of an ordinary charged dielectric. Thus when the separation occurs, there are charged combinations of this kind in each portion. Consequently, each piece has the same charge characteristics as the original unbroken object.

It was pointed out in Volume I that the existence of positive* and negative* charges in close proximity, as required by the nuclear theory of the atom, is incompatible with the observed behavior of charges of opposite polarity. These observations show that such charges neutralize each other long before they reach separations as small as those which would exist in the hypothetical nuclear atom. This is a decisive argument against the validity of the nuclear theory. It is appropriate, therefore, to note that the existence of both positive* and negative* charges in objects under the influence of inducing charges does not conflict with our finding that there is a minimum distance (identified as the *natural unit *of distance, 4.56×10^{-6}cm) within which charges of opposite polarity cannot coexist. The coexistence of *induced *positive and negative charges is possible because they are *forcibly prevented *from reaching the limiting distance at which they would combine. If the external charge is removed, the induced charges *do *combine and neutralize each other.

In charging by induction it is often convenient to make use of *grounding*, which is simply connecting the inductively charged object to the earth by means of a conductor. The earth is electrically neutral, and so large that it is insensitive to gains or losses of charge in the amounts actually encountered in practice. If object Y is grounded while under the influence of a negative* inducing charge, the negatively* charged electrons on the far side of the object are forced through the conductor into the earth. Breaking the ground connection then leaves only positive* charges on object Y, and this object remains positively* charged after the object X that contains the inducing charge is removed. If the induction process is initiated by a positive* charge on object X, the ground connection permits electrons to be pulled from the earth and charged negatively* to neutralize the positive* charges on Y, leaving only negative* charges. Breaking the ground connection then leaves Y negatively* charged.

The locations occupied by the charges in any charged conducting object not subject to inductive forces are determined by the repulsion between the charges, which operates to produce the maximum separation. If the object is under the influence of outside charges, the charge locations are determined by the net effect of the inductive potential and the repulsion between like charges. In either case, the result is that the charges are confined to the outside surfaces of the conducting materials, and, except for local variations in very irregular bodies, there are no charges in the interiors. The same considerations apply if the objects are hollow. The inside walls of such objects carry no charges. These walls may be charged by placing an insulated charged object in the hollow interior, but in that case the inside walls are “outside” from the standpoint of the inducing charge; that is, they are the locations closest to that charge.

The observed concentration of charge at the conductor surfaces is another direct contradiction of the accepted theory of the electric current, which views the current as a movement of charges. This concentration at the surface is due to the mutual repulsion between the charges, which drives them to opposite sides of the conductor. The repulsive force is not altered if the charges move along the conductor, since the direction of this force is perpendicular to the direction of movement. Nor would the presence of positive* charges on the interior atoms of the conductor alter this situation, if any such charges existed. If the electrons, or any portion of them, are firmly held by the attraction of the hypothetical proton charges, then they cannot move as an electric current. If they are free to move in response to an electric potential gradient, then they are also free to move to the surfaces of the conductor under the influence of their mutual repulsions.

From this it follows that if present-day electric theory were correct the current should flow only along the outer surfaces of the conductors. In fact, however, electric resistance is generally proportional to the cross-sectional area of the conductor, indicating that the motion takes place fairly uniformly throughout the entire cross section. This adds one more item of evidence supporting the finding that the electric current is a movement of uncharged electrons, not of charges.

Since no charges are induced within a hollow conductor by an outside charge, any object within a conducting enclosure is insulated against the effects of an electric charge. Similar elimination, or reduction, of these effects is accomplished by conductors of other shapes that are interposed between the charge and the objects under consideration. This is the process known as * shielding, *which has a wide variety of applications in electrical practice.

Within the limits to which the present examination of electrical phenomena has been carried, there does not appear to be any major error in the conventional dimensional assignments, other than those discussed in the preceding pages. Aside from the errors that have been identified, the SI system is dimensionally consistent, and consistent with the mechanical system of quantities. The space-time dimensions of the most commonly used electrical units are listed in Table 28. The first column of this table lists the symbols that are used in this work. The other columns are self-explanatory.

The natural units of most of these quantities can be derived from the natural units previously evaluated. Those of the remaining quantities can be calculated by the methods used in the previous determinations, but the evaluation is complicated by the fact that the measurement systems in current use are not internally consistent, and it is not possible to identify a constant numerical value that relates any one of these systems to the natural system of units, as was done for the mechanical quantities that involve the unit of mass. Neither the SI system nor the cgs system of electrical units qualifies as a single measurement system in this sense. Both are combinations of systems. In the present discussion we will distinguish the measurement systems, as here defined, by the numerical coefficients that apply, in these systems, to the natural unit of space, s, and inverse speed, t/s.

On the basis of the values of the natural units of space and time in cgs terms established in Volume I, the numerical coefficient of the natural unit of s, regardless of what name is applied to the quantity, should be 4.558816×10^{-6}, while that of the natural unit of t/s should be 3.335635×10^{-11}. In the mechanical system of measurement the quantity s is identified in its most general character as space, and the unit has the appropriate numerical coefficient. But the concept of mass was introduced into the t/s quantity, here called energy, and an arbitrary mass unit was defined. This had the effect of modifying the numerical values of the natural units of energy and its derivatives by the factor 4.472162×10^{7}, as explained in Volume I.

The definition of the unit of charge (esu) by means of the Coulomb equation in the electrostatic system of measurement was originally intended as a means of fitting the electrical quantities into the mechanical measurement system. But, as pointed out in Chapter14, there is an error in the dimensional assignments in that equation which introduces a deviation from the mechanical values. The electrostatic unit of charge and the other electric units that incorporate the esu therefore constitute a separate system of measurement, in which t/s is identified with electric charge. The unit of this quantity was evaluated from the Faraday constant in Chapter 9 as 4.80287×10^{-10} esu.

Unit charge can also be measured directly, inasmuch as some physical entities are incapable of taking more than one unit of electric charge. The charge of the electron, for instance, is one unit. Direct measurement of this charge is somewhat more difficult than the derivation of the natural unit from the Faraday constant, but the direct measurements are in reasonably good agreement with the indirectly derived value. In fact, as noted in Chapter 14, clarification of the small scale factors that affect these phenomena will probably bring all values, including the one that we have derived theoretically, into agreement.

An electromagnetic unit (emu) analogous to the esu can be obtained by magnetic measurements, and this forms the basis of an electromagnetic system of measurement. The justification for using the emu as a unit of electrical measurement is provided by the assumption that it is an electric unit derived from an electromagnetic process. We now find, however, this it is, in fact, a magnetic unit; that is, it is a two-dimensional unit. It is therefore a unit of t^{2}/s^{2} rather than a unit of t/s. To obtain an electric (one-dimensional) unit, t/s, corresponding to the esu from the emu it is necessary to multiply the measured value of the emu coefficient, 1.602062×10^{-20} by the natural unit of s/t, 2.99793×10^{10} cm/sec. This brings us back to the electrostatic unit, 4.80287×10^{-10}esu. The electromagnetic system is thus nothing more than the electrostatic system to which an additional factor, meaningless in the electrical context, has been applied.

The SI system of units is a modification of the electromagnetic system. In the early days of electrical measurement the ampere was selected as the fundamental unit, and was defined on an arbitrary basis. After more information had been accumulated, and the desirability of relating the measurement system to physical fundamentals was recognized, the electromagnetic (emu) system was adopted for general use, but in order to avoid making a radical change in the size of the ampere, an arbitrary factor of 10 was introduced. As M. McCaig remarks, the appearance of such a number “in a primary definition is unusual; it arises because although this definition is intended to fix the value of the ampere, we have already decided in advance fairly precisely the value we desire the unit to have.”^{59}

The arbitrary modification of the emu values changed the numerical coefficient of the natural unit of t/s to 1.602062×10^{-19}. Because of the lack of distinction between electric charge (t/s) and electric quantity (s) in current practice, the same unit is used for both of these physical quantities in all three of the electrical measurement systems, as shown in Table 29.

s | t/s | |
---|---|---|

Space-time (cgs) | 4.558816×10^{-6} | 3.335635×10^{-11} |

Mechanical | 4.558816×10^{-6} | 1.49175×10^{-3} |

Electrostatic | 4.80287×10^{-10} | 4.80287×10^{-10} |

Electromagnetic | 1.602062×10^{-20} | 1.602062×10^{-20} |

SI modification | 1.602062 ×10^{-19} | 1.602062×10^{-19} |

In applying the principle of the equivalence of natural units to electrical quantities it is necessary to take into account these differences between the numerical values applying to the different systems. For example, the natural unit of capacitance, the quantity that plays the principal role in the phenomena discussed in Chapter 15, is the natural unit of electric charge divided by the natural unit of voltage, t/s × s^{2}/t = s. On the basis of the explanation of the natural electrical units given in the preceding paragraphs, the value of the natural unit of electric charge in the cgs electrostatic system is 4.80287×10^{-10} esu. The natural unit of capacitance is this value divided by the natural unit of voltage, which was evaluated in Chapter 9 as 9.31146×10^{8} volts. The result is 5.15802×10^{-18} farads. As we found earlier, the farad is a unit of space. The natural unit of space derived in Volume I is 4.558816×10^{-6} cm. Dividing these two values, we obtain 1.1314×10^{-12} as the ratio of the numerical coefficients of the natural units. From geometric measurements, the centimeter, as a unit of capacitance, has been found equal to 1.11126×10^{-12} farads. The theoretical and experimental values are therefore in agreement within the limits of accuracy to which the present study of the electric relations has been carried.

In this case the application of the equivalence principle merely corroborates an experimental result. Its value as an investigative tool derives from the fact that it is equally applicable in situations where nothing is available from other sources.