Chapter V                               THE FUNDAMENTAL LAWS OF ELECTRODYNAMICS                    



Page 146

     "The fact that a certain kind of ore, magnetite, attracts iron, and that rubbed amber (elektron in Greek) attracts and holds light bodies was already known to the ancients. But the science of magnetism and electricity are products of more recent times which had been trained by Galileo and Newton to ask rational questions of nature with the help of experiment.
    The fundamental facts of electrical phenomena, which we shall now recapitulate briefly, were established after the year 1600. At that time friction was the exclusive means of producing electrical effects. Gray discovered (1729) that metals, when brought into contact with bodies that had been electrified by friction, themselves acquire similar properties. He showed that electricity can be conducted in metals. This led to the classification of substances as conductors and nonconductors (insulators) It was discovered by du Fay (1730) that electrical action is not always attraction but may also be repulsion
To account for this fact he assumed the existence of two fluids (nowadays we call them positive and negative electricity), and he established that similarly charged bodies repel each other, while oppositely charged bodies attract each other.
     We shall define the concept of electric charge quantitatively. In doing so we will not follow the oftentimes very circuitous steps of argument that led historically to the enunciation of the concepts and laws, but rather we shall select a series of definitions and experiments in which the logical sequence emerges most clearly.
      Let us imagine a body M that has some how been electrified by friction. This now acts attractively or repulsively on other electrified  

/ Page 147  /  

bodies. To study this action we shall take small test bodies, say spheres, whose diameters are very small compared with the distance of their closest approach to the body M. If we bring a test body P near the body M, P experiences a statistical force of definite magnitude and direction which may be measured by the methods of mechanics, say, by balancing it against a weight with the help of levers and threads. It is found qualitevely that the force decreases with increasing distance P M.
     We next take two such test bodies P1 and P 2, bring them in turn to the same point in the vicinity of M, and measure in each case the forces K1 and K2 as regards size and direction. We shall henceforth adopt the convention that opposite forces are to be regarded as being in the same direction and having opposite signs. Experiment shows that the two forces have the same direction but that their values may have different signs.
      Now let us bring the two test bodies to a different point near M and let us again measure the forces K1' and K2' as regards value and direction. Again they have the same direction, but in general they have different values and different signs

                                                       K1 = K1'.
                                                       K2    K2'
From this result we may conclude:
       1.  The direction of the force exerted by an electrified body M on a small test body P does not depend at all on the nature and the amount of electrification of the test body, but only on the properties of the body M
       2  The ratio of the forces exerted on two test bodies brought to the same point in turn is quite independent of the choice of the point, that is, of the position, nature, and electrification of the body M. It depends only on the properties of the test bodies.
       We now choose a definite test body, electrified in a definite way, and let its charge be the unit of charge or amount of electricity q. With the aid of this test body we measure the force that the body M exerts at many places.  this force be denoted by Kq.  Then  

/ Page 148   /

this also determines the direction of the force K exerted on any other test body p. The ratio K: Kq, however, depends only on the test body P and defines the ratio e of the electric charge of P  and the unit of charge q. This may be positive or negative depending upon whether K and Kq are in the same or in opposite directions. Thus we have in any position:  K = Kq .
                E      q

From this one concludes that K depends only on the electrical nature of the body M.
                                                e
                       Therefore we call the quotient  K = Kq  the electrical field strength E. This quantity E determines the electrical
                                                                          e      q
  
action of M in the surrounding space, or as we usually say, its electric field. From K= E follows
                                                                                                                                    e

                                                                                                                                K = e E.                                                     (45)

The scribe puzzling, enquired as to the inclusion of diagrams and formula references supplied by good brother Born.
On advice from the only one who knows, Zed Aliz Zed,a being guided towards right action. asked the scribe to omit those of the suggested formulae and diagram references that had to be omitted and to include any that ought to be included betwixt and between pages 146 and 224 of Brother Born's work.
Not altogether puzzled by such an answer as that, the far yonder scribe would do just that, to the best of that's scribes abilities. Nevertheless, understanding its importance within the creator schema of rings, the scribe wondered aloud az to the Zed Aliz Zed's hieroglyphics conundrum, and not believing in working piecemeal had a mite to eat.  
After which the Zed Aliz Zed The Far Yonder Scribe, the shadows on my shoulder and attendant mirror images re-affirmed their golden thread, and did each enter, as spiritual glow worms, the cave of the Minotaur.

Page 148 continued  

"As for the choice of unit charge, it would be almost impossible to fix this in a practical way by a decree concerning the electrification of a definite test body; a mechanical definition would be preferable. This can be arrived at as follows:
    We first give two test bodies equal charges. The criterion of equal charges is that they are subject to the same force from the same body M when placed at the same point near M. The two bodies will then repel each other with the same force. We now say that their charge equals the unit of charge q if this repulsion is equal to the unit of force when the distance between the two test bodies is equal to unit length. No assumption is made here about the dependence of the force on the distance.
      Through these definitions the amount of electricity or the electric charge becomes a measurable quantity just as length, mass or force may be measured.
      The most important law about amounts of electricity, which was enunciated independently in 1747 by Watson and Franklin, is that in every electrical process equal ammounts of positive and negative electricity are always formed. For example, if we rub a glass rod with a piece of silk, the glass rod becomes charged with positive electricity; an exactly equal negative charge is then found on the silk.  

/ Page 149  1 x 4 x 9 = 36    3 + 6 = 9   /  

This empirical fact may be interpreted by saying that the two kinds of electrification are not generated by friction but are only separated. They may be thought of as two fluids that are present in all bodies in equal quantities. In nonelectrified "neutral " bodies they are every-where present to the same amount so that their outward effects are counterbalanced. In electrified bodies they are separated. One part of the positive electricity, say, has flowed from one body to another; just as much negative has flowed in the reverse direction.
      But it is clearly sufficient to assume one fluid that can flow inde-pendently of matter. The we must ascribe to matter that is free of this fluid a definite charge say  positive, and to the fluid the opposite charge, that is, negative. Electrification consists of the flowing of negative fluid from one body to the other. The first body will then become positive because the positive charge of the matter is no longer wholly compensated; the other becomes negative because it has an excess of negative fluid.
      The struggle between the supporters of these two hypotheses, the one-fluid theory and the two fluid theory, lasted a long time, and of course remained futile and purposeless until it was decided by the discovery of new facts. We shall not enter further into these discussions, but shall only state briefly that characteristic differences were finally found in the behaviour of the two kinds of electricity; these differences indicated that positive electrification is actually firmly attached to matter but that negative electrification can move more or less freely. This doctrine still holds today. We shall revert to this point later in dealing with the theory of electrons.
                                         Another controversy arose  around the question of how the electri-cal forces of attraction and repulsion are transmitted                
through space. The first decades of electrical research came before the Newtonian theory of attraction. Action at a distance seemed unthinkable. Metaphysical theorems were held to be valid (for example, that matter can act only at points where it is present) and diverse hypotheses were evolved to explain electrical forces - for example, that emanations flowed from the charged bodies and exerted a pressure when they impinged on bodies, and similar assumptions. But after Newton's theory of gravitation had been established, the idea of a force acting directly at a distance gradually became a habit of thought. For it is, indeed, nothing more than a thought habit when an idea impresses  

/ Page 150  /

itself so bly on minds that it is used as the ultimate principal of explanation. It does not then take long for metaphysical speculation, often in the garb of philosophic criticism to maintain that the correct or accepted principle of explanation is a logical necessity and that its opposite cannot be imagined. But fortunately progressive empirical science does not as a rule trouble about this, and when new facts demand it, it often has recourse to ideas that have been condemned. The development of the doctrine of electric and magnetic forces is an example of such a cycle of theories. First came a theory of contiguous action based on metaphysical grounds, later a theory of action at a distance on Newton's model. Finally this became transformed, owing to the discovery of new facts, into a general theory of contiguous action again. The fluctuation is no sign of weakness. For it is not the pictures that are connected with the theories which are the essential features but the empirical facts and their conceptual relationships. Yet if we follow these we see no fluctuation but only a continuous development full of inner logical consistency. We may justifiably pass by the first theoretical attempts of pre-Newtonian times because the facts were known too incompletely to furnish really convincing starting points. But the rise of the theory of action at a distance in Newtonian mechanics is founded quite solidly on facts of observation. Research which had at its disposal only the experimental means of the eighteenth century was bound to come to the decision that the electric and magnetic forces act at a distance in the same way as gravitation. Even nowadays it is still permissible, from the point of view of the highly developed theories of contiguous action of Faraday and Maxwell, to represent electro- and magnetostatic forces by means of actions at a distance, and when properly used they lead to correct results.
      The idea that electric forces act like gravitation at a distance was first conceived by Aepinus (1759). He did not succeed in setting up the correct law for the dependence of electric actions on the dis-tance, but he was able to explain the phenomenon of electrostatic induction qualitively. This consists of a charged body acting attractively not only on other charged bodies but also on uncharged bodies, particularly on conducting bodies: a charge of the opposite sign is induced on the side of the influenced body nearest the acting body, whereas a charge of the same sign is driven to the farther side ( Fig 78 );  

/ Page 151  /  

hence, since the forces decrease with increasing distance, the attraction outweighs the repulsion.
     The exact law of this decrease was presumably first found by Priestly, the discoverer of oxygen (1767). He discovered the law in an ingenious indirect way which was more convincing than a direct measurement would have been. Independently Cavendish (1771) derived the law by similar  reasoning. But it received its name from the physicist who first proved it by measuring the forces directly, Coulomb (1785)..."
 
            Fig. 78  A charged body M in-fluences charges             Fig. 79 Derivation of Cou-lomb's law.
                      on an originally uncharged body



  "The argument of Priestly and Cavendish ran somewhat as follows: If an electric charge is given to a conductor, then it cannot remain in equilibrium in the interior of the conducting substance, since particles of the same charge repel each other. Rather, they must tend to the outer surface where they distribute themselves in a certain way so as to be in equilibrium. Now experiment teaches very definitely that no electric field exists within a space that is enclosed on all sides by metallic walls, no matter how bly the envelope is charged The charges on the outer surface of the empty space must thus distribute themselves so that the force exerted at each point in the interior vanishes. Now, if the empty space has the particular form of a sphere, the charge for reasons of symmetry, can only be distributed uniformly over the surface..."

Page 153  

"...In conformity with our convention about the unit of electric charge we must set C= 1 x unit of forcex (unit of length) 2; then we define the dimensions of charge by putting C=q2. Now the force between two unit charges a unit distance apart is to be equal to one unit of force. With this convention the force that two bodies carrying charges e1 and e2 and at a distance r apart exert on each other is
                                    K=e1e2 .                                                                                                                                               (46)
                                            r2

This is Coulombs law. In its formulation we assume, of course, that the greatest diameter of the charged bodies is small compared with their distances apart. This restriction means that we have to do, just as in the case of gravitation, with an idealized elementary law. To deduce from it the action of bodies of finite extent we must consider the electricity distributed over them to be divided into small parts, then calculate the effects of all the particles of the one body on all those of the others in pairs and sum them."

Page 154    

"After Coulomb's law had been established, electrostatics became a mathematical science. Its most important problem is this: Given the total quantity of electricity on conducting bodies, to calculate the distribution of charges on them under the action of their mutual influence, and also the forces due to these charges. The develop-ment of this mathematical problem is interesting in that it very soon became changed from the original formulation based on the theory of action at a distance to a theory of pseudocontiguous action, that is, in place of the summations of Coulomb forces there were obtained differential equations in which the field E or a related quantity called potential occurred as the unknown. However we cannot discuss these purely mathematical questions any further here but only mention the names of Laplace (1782), Poisson (1813), and Gauss (1840) who have played a prominent role in their solution. We shall emphasize only one point. In this treatment of electrostatics, which is usually called the theory of potential, we are not dealing with a true theory of contiguous action in the sense which we attached to this expression before..." "...for the differential equations refer only to the change in the intensity of field from place to place and contains no term that expresses a change in time. Hence they entail no transmission of electric force with finite velocity but, in spite of their differential form, they represent an instantaneous action at a distance.
    The theory of magnetism was developed in the same way as that of electrostatics. We may, therefore express ourselves briefly.
    A lozenge-shaped magnetized body, a magnet needle, has two poles, that is, point from which the magnetic force seems to start out, and the law holds that like poles repel, unlike poles attract one another. If we break a magnet in half, the two parts do not carry opposite magnetic charges, but each part shows a new pole near the new surface and again represents a complete magnet with two equal but opposite poles. This holds, no matter into how many parts the magnet be broken.
   From this it has been concluded that there are indeed two kinds of magnetism as in the case of electricity except that they cannot move freely, and that they are present in the smallest particles of matter, molecules, in equal quantities, but separated by a small distance. Thus each molecule is itself a small magnet with a north and a south  

/ Page 155   /

pole (Fig. 80). In a body that is not magnetized all the elementary magnets are in complete disorder. Magnetization consists of bring-ing them into the same direction. Then the effects of the alternate north (+) and south (- ) poles counterbalance, except at the two ends which therefore seem to be the sources of the magnetic effects..."
 
                                         Fig 80 A magnetized body consisting of elementary magnets.
 
     "... By using a very long, thin magnetized needle one can be sure that in the vicinity of the one pole the force of the other becomes negligible. Hence in magnetism, too we may operate with test bodies, namely with the poles of very long, thin magnetic rods. These allow us to carry out all the measurements that we have already discussed in the case of electricity..."      "...Clearly the dimensions of magnetic quantities are the same as those of the corresponding electric quantities, and their units have the same notation in the C.G.S. system.
    The mathematical theory of magnetism runs almost parallel with that of electricity. The most essential difference is that magnetism is attached to the molecules, and that the measureable accumulations 

 / Page 156  /

that condition the occurrence of poles in the case of finite magnets arise only owing to the summation of molecules that point in the same direction. One cannot separate the two kinds of magnetism and make a body, for example a north pole.