A Treatise on Electricity and Magnetism/Part IV/Chapter XXIII

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On the Explanation of Ampère's Formula given by Gauss and Weber.

846.] The attraction between the elements and of two circuits, carrying electric currents of intensity and is, by, Ampère's formula,


or (2)

the currents being estimated in electromagnetic units. See Art. 526.

The quantities, whose meaning as they appear in these expressions we have now to interpret, are

and the most obvious phenomenon in which to seek for an interpretation founded on a direct relation between the currents is the relative velocity of the electricity in the two elements.

847.] Let us therefore consider the relative motion of two particles, moving with constant velocities and along the elements and respectively. The square of the relative velocity of these particles is


and if we denote by the distance between the particles,




where the symbol indicates that, in the quantity differentiated, the coordinates of the particles are to be expressed in terms of the time.

It appears, therefore, that the terms involving the product in the equations (3), (5), and (6) contain the quantities occurring in (1) and (2) which we have to interpret. We therefore endeavour to express (1) and (2) in terms of , and . But in order to do so we must get rid of the first and third terms of each of these expressions, for they involve quantities which do not appear in the formula of Ampère. Hence we cannot explain the electric current as a transfer of electricity in one direction only, but we must combine two opposite streams in each current, so that the combined effect of the terms involving and may be zero.

Fechner's Hypothesis.

848.] Let us therefore suppose that in the first element, , we have one electric particle, , moving with velocity and another, and moving with velocity and in the same way two particles, and , in moving with velocities and respectively.

The term involving for the combined action of these particles is


Similarly (8)

and (9)

In order that may be zero, we must have either


According to Fechner's hypothesis, the electric current consists of a current of positive electricity in the positive direction, combined with a current of negative electricity in the negative direction, the two currents being exactly equal in numerical magnitude, both as respects the quantity of electricity in motion and the velocity with which it is moving. Hence both the conditions of (10) are satisfied by Fechner's hypothesis.

But it is sufficient for our purpose to assume, either―

That the quantity of positive electricity in each element is numerically equal to the quantity of negative electricity; or―

That the quantities of the two kinds of electricity are inversely as the squares of their velocities.

Now we know that by charging the second conducting wire as a whole, we can make either positive or negative. Such a charged wire, even without a current, according to this formula, would act on the first wire carrying a current in which has a value differing from zero. Such an action has never been observed.

Therefore, since the quantity may be shewn experimentally not to be always zero, and since the quantity is not capable of being experimentally tested, it is better for these speculations to assume that it is the latter quantity which invariably vanishes.

849.] Whatever hypothesis we adopt, there can be no doubt that the total transfer of electricity, reckoned algebraically, along the first circuit, is represented by

where is the number of units of statical electricity which are transmitted by the unit electric current in the unit of time, so that we may write equation (9)


Hence the sums of the four values of (3), (5), and (6) become




and we may write the two expressions (1) and (2) for the attraction between and


and (16)

Action At A Distance.

850.] The ordinary expression, in the theory of statical electricity, for the repulsion of two electrical particles and is and


which gives the electrostatic repulsion between the two elements if they are charged as wholes.

Hence, if we assume for the repulsion of the two particles either of the modified expressions


or (19)

we may deduce from them both the ordinary electrostatic forces, and the forces acting between currents as determined by Ampère.

Formulae of Gauss and Weber.

851.] The first of these expressions, (18), was discovered by Gauss[1] in July 1835, and interpreted by him as a fundamental law of electrical action, that 'Two elements of electricity in a state of relative motion attract or repel one another, but not in the same way as if they are in a state of relative rest.' This discovery was not, so far as I know, published in the lifetime of Gauss, so that the second expression, which was discovered independently by W.Weber, and published in the first part of his celebrated Elektrodynamische Maasbestimmungen[2] , was the first result of the kind made known to the scientific world.

852.] The two expressions lead to precisely the same result when they are applied to the determination of the mechanical force between two electric currents, and this result is identical with that of Ampère. But when they are considered as expressions of the physical law of the action between two electrical particles, we are led to enquire whether they are consistent with other known facts of nature.

Both of these expressions involve the relative velocity of the particles. Now, in establishing by mathematical reasoning the well-known principle of the conservation of energy, it is generally assumed that the force acting between two particles is a function of the distance only, and it is commonly stated that if it is a function of anything else, such as the time, or the velocity of the particles, the proof would not hold.

Hence a law of electrical action, involving the velocity of the particles, has sometimes been supposed to be inconsistent with the principle of the conservation of energy.

853.] The formula of Gauss is inconsistent with this principle, and must therefore be abandoned, as it leads to the conclusion that energy might be indefinitely generated in a finite system by physical means. This objection does not apply to the formula of Weber, for he has shewn[3] that if we assume as the potential energy of a system consisting of two electric particles,


the repulsion between them, which is found by differentiating this quantity with respect to r, and changing the sign, is that given by the formula (19). Hence the work done on a moving particle by the repulsion of a fixed particle is , where and are the values of at the beginning and at the end of its path. Now depends only on the distance, , and on the velocity resolved in the direction of . If, therefore, the particle describes any closed path, so that its position, velocity, and direction of motion are the same at the end as at the beginning, will be equal to , and no work will be done on the whole during the cycle of operations.

Hence an indefinite amount of work cannot be generated by a particle moving in a periodic manner under the action of the force assumed by Weber.

854.] But Helmholtz, in his very powerful memoir on the 'Equations of Motion of Electricity in Conductors at Rest' [4], while he shews that Weber's formula is not inconsistent with the principle of the conservation of energy, as regards only the work done during a complete cyclical operation, points out that it leads to the conclusion, that two electrified particles, which move according to Weber's law, may have at first finite velocities, and yet, while still at a finite distance from each other, they may acquire an infinite kinetic energy, and may perform an infinite amount of work.

To this Weber[5] replies, that the initial relative velocity of the particles in Helmholtz's example, though finite, is greater than the velocity of light; and that the distance at which the kinetic energy becomes infinite, though finite, is smaller than any magnitude which we can perceive, so that it may be physically impossible to bring two molecules so near together. The example, therefore, cannot be tested by any experimental method.

Helmholtz[6] has therefore stated a case in which the distances are not too small, nor the velocities too great, for experimental verification. A fixed non-conducting spherical surface, of radius , is uniformly charged with electricity to the surface-density . A particle, of mass and carrying a charge of electricity, moves within the sphere with velocity . The electrodynamic potential calculated from the formula (20) is


and is independent of the position of the particle within the sphere. Adding to this , the remainder of the potential energy arising from the action of other forces, and , the kinetic energy of the particle, we find as the equation of energy


Since the second term of the coefficient of may be increased in definitely by increasing , the radius of the sphere, while the surface-density remains constant, the coefficient of may be made negative. Acceleration of the motion of the particle would then correspond to diminution of its vis viva, and a body moving in a closed path and acted on by a force like friction, always opposite in direction to its motion, would continually increase in velocity, and that without limit. This impossible result is a necessary consequence of assuming any formula for the potential which introduces negative terms into the coefficient of .

855.] But we have now to consider the application of Weber's theory to phenomena which can be realized. We have seen how it gives Ampère's expression for the force of attraction between two elements of electric currents. The potential of one of these elements on the other is found by taking the sum of the values of the potential for the four combinations of the positive and negative currents in the two elements. The result is, by equation (20), taking the sum of the four values of ,


and the potential of one closed current on another is



In the case of closed currents, this expression agrees with that which we have already (Art. 524) obtained[7].

Weber's Theory of the Induction of Electric Currents.

856.] After deducing from Ampère's formula for the action between the elements of currents, his own formula for the action between moving electric particles, Weber proceeded to apply his formula to the explanation of the production of electric currents by magneto-electric induction. In this he was eminently successful, and we shall indicate the method by which the laws of induced currents may be deduced from Weber s formula. But we must observe, that the circumstance that a law deduced from the pheno mena discovered by Ampère is able also to account for the phenomena afterwards discovered by Faraday does not give so much additional weight to the evidence for the physical truth of the law as we might at first suppose.

For it has been shewn by Helmholtz and Thomson (see Art. 543), that if the phenomena of Ampère are true, and if the principle of the conservation of energy is admitted, then the phenomena of induction discovered by Faraday follow of necessity. Now Weber's law, with the various assumptions about the nature of electric currents which it involves, leads by mathematical transformations to the formula of Ampère. Weber's law is also consistent with the principle of the conservation of energy in so far that a potential exists, and this is all that is required for the application of the principle by Helmholtz and Thomson. Hence we may assert, even before making any calculations on the subject, that Weber's law will explain the induction of electric currents. The fact, therefore, that it is found by calculation to explain the induction of currents, leaves the evidence for the physical truth of the law exactly where it was.

On the other hand, the formula of Gauss, though it explains the phenomena of the attraction of currents, is inconsistent with the principle of the conservation of energy, and therefore we cannot assert that it will explain all the phenomena of induction. In fact, it fails to do so, as we shall see in Art. 859.

857.] We must now consider the electromotive force tending to produce a current in the element due to the current in , when is in motion, and when the current in it is variable.

According to Weber, the action on the material of the conductor of which is an element, is the sum of all the actions on the electricity which it carries. The electromotive force, on the other hand, on the electricity in is the difference of the electric forces acting on the positive and the negative electricity within it. Since all these forces act in the line joining the elements, the electromotive force on is also in this line, and in order to obtain the electromotive force in the direction of we must resolve the force in that direction. To apply Weber's formula, we must calculate the various terms which occur in it, on the supposition that the element is in motion relatively to , and that the currents in both elements vary with the time. The expressions thus found will contain terms involving , , , , , and terms not involving or , all of which are multiplied by . Examining, as we did before, the four values of each term, and considering first the mechanical force which arises from the sum of the four values, we find that the only term which we must take into account is that involving the product .

If we then consider the force tending to produce a current in the second element, arising from the difference of the action of the first element on the positive and the negative electricity of the second element, we find that the only term which we have to examine is that which involves . We may write the four terms included in , thus

Since , the mechanical force arising from these terms is zero, but the electromotive force acting on the positive electricity is and that acting on the negative electricity is equal and opposite to this.

858.] Let us now suppose that the first element is moving relatively to with velocity in a certain direction, and let us denote by and , the angle between the direction of and that of and of respectively, then the square of the relative velocity, , of two electric particles is


The term in is the same as in equation (3). That in , on which the electromotive force depends, is

We have also for the value of the time-variation of in this case


where refers to the motion of the electric particles, and to that of the material conductor. If we form the square of this quantity, the term involving , on which the mechanical force depends, is the same as before, in equation (5), and that involving , on which the electromotive force depends, is

Differentiating (26) with respect to , we find


We find that the term involving is the same as before in (6).

The term whose sign alters with that of is .

859.] If we now calculate by the formula of Gauss (equation (18)), the resultant electrical force in the direction of the second element arising from the action of the first element , we obtain


As in this expression there is no term involving the rate of variation of the current , and since we know that the variation of the primary current produces an inductive action on the secondary circuit, we cannot accept the formula of Gauss as a true expression of the action between electric particles.

860.] If, however, we employ the formula of Weber, (19), we obtain


or (30)

If we integrate this expression with respect to and , we obtain for the electromotive force on the second circuit


Now, when the first circuit is closed,

Hence (32)

But (33)

Hence we may write the electromotive force on the second circuit


which agrees with what we have already established by experiment; Art. 539.

On Weber's Formula, considered as resulting from an Action transmitted from one Electric Particle to the other with a Constant Velocity.


861.] In a very interesting letter of Gauss to W. Weber[8] he refers to the electrodynamic speculations with which he had been occupied long before, and which he would have published if he could then have established that which he considered the real keystone of electrodynamics, namely, the deduction of the force acting between electric particles in motion from the consideration of an action between them, not instantaneous, but propagated in time, in a similar manner to that of light. He had not succeeded in making this deduction when he gave up his electrodynamic researches, and he had a subjective conviction that it would be necessary in the first place to form a consistent representation of the manner in which the propagation takes place.

Three eminent mathematicians have endeavoured to supply this keystone of electrodynamics.

862.] In a memoir presented to the Royal Society of Gottingen in 1858, but afterwards withdrawn, and only published in Poggendorff's Annalen in 1867, after the death of the author, Bernhard Riemann deduces the phenomena of the induction of electric currents from a modified form of Poisson's equation

where is the electrostatic potential, and a velocity.

This equation is of the same form as those which express the propagation of waves and other disturbances in elastic media. The author, however, seems to avoid making explicit mention of any medium through which the propagation takes place.

The mathematical investigation given by Riemann has been examined by Clausius[9], who does not admit the soundness of the mathematical processes, and shews that the hypothesis that potential is propagated like light does not lead either to the formula of Weber, or to the known laws of electrodynamics.

863.] Clausius has also examined a far more elaborate investigation by C. Neumann on the 'Principles of Electrodynamics' [10]. Neumann, however, has pointed out[11] that his theory of the transmission of potential from one electric particle to another is quite different from that proposed by Gauss, adopted by Riemann, and criticized by Clausius, in which the propagation is like that of light. There is, on the contrary, the greatest possible difference between the transmission of potential, according to Neumann, and the propagation of light.

A luminous body sends forth light in all directions, the intensity of which depends on the luminous body alone, and not on the presence of the body which is enlightened by it.

An electric particle, on the other hand, sends forth a potential, the value of which, , depends not only on , the emitting particle, but on the receiving particle, and on the distance between the particles at the instant of emission.

In the case of light the intensity diminishes as the light is propagated further from the luminous body; the emitted potential flows to the body on which it acts without the slightest alteration of its original value.

The light received by the illuminated body is in general only a fraction of that which falls on it; the potential as received by the attracted body is identical with, or equal to, the potential which arrives at it.

Besides this, the velocity of transmission of the potential is not, like that of light, constant relative to the æther or to space, but rather like that of a projectile, constant relative to the velocity of the emitting particle at the instant of emission.

It appears, therefore, that in order to understand the theory of Neumann, we must form a very different representation of the process of the transmission of potential from that to which we have been accustomed in considering the propagation of light. Whether it can ever be accepted as the 'construirbar Vorstellung' of the process of transmission, which appeared necessary to Gauss, I cannot say, but I have not myself been able to construct a consistent mental representation of Neumann's theory.

864.] Professor Betti[12], of Pisa, has treated the subject in a different way. He supposes the closed circuits in which the electric currents flow to consist of elements each of which is polarized periodically, that is, at equidistant intervals of time. These polarized elements act on one another as if they were little magnets whose axes are in the direction of the tangent to the circuits. The periodic time of this polarization is the same in all electric circuits. Betti supposes the action of one polarized element on an other at a distance to take place, not instantaneously, but after a time proportional to the distance between the elements. In this way he obtains expressions for the action of one electric circuit on another, which coincide with those which are known to be true. Clausius, however, has, in this case also, criticized some parts of the mathematical calculations into which we shall not here enter.

865.] There appears to be, in the minds of these eminent men, some prejudice, or à priori objection, against the hypothesis of a medium in which the phenomena of radiation of light and heat, and the electric actions at a distance take place. It is true that at one time those who speculated as to the causes of physical phenomena, were in the habit of accounting for each kind of action at a distance by means of a special æthereal fluid, whose function and property it was to produce these actions. They filled all space three and four times over with æthers of different kinds, the properties of which were invented merely to 'save appearances' , so that more rational enquirers were willing rather to accept not only Newton's definite law of attraction at a distance, but even the dogma of Cotes[13], that action at a distance is one of the primary properties of matter, and that no explanation can be more intelligible than this fact. Hence the undulatory theory of light has met with much opposition, directed not against its failure to explain the phenomena, but against its assumption of the existence of a medium in which light is propagated.


866.] We have seen that the mathematical expressions for electrodynamic action led, in the mind of Gauss, to the conviction that a theory of the propagation of electric action in time would be found to be the very key-stone of electrodynamics. Now we are unable to conceive of propagation in time, except either as the flight of a material substance through space, or as the propagation of a condition of motion or stress in a medium already existing in space. In the theory of Neumann, the mathematical conception called Potential, which we are unable to conceive as a material substance, is supposed to be projected from one particle to another, in a manner which is quite independent of a medium, and which, as Neumann has himself pointed out, is extremely different from that of the propagation of light. In the theories of Riemann and Betti it would appear that the action is supposed to be propagated in a manner somewhat more similar to that of light.

But in all of these theories the question naturally occurs:―If something is transmitted from one particle to another at a distance, what is its condition after it has left the one particle and before it has reached the other? If this something is the potential energy of the two particles, as in Neumann's theory, how are we to conceive this energy as existing in a point of space, coinciding neither with the one particle nor with the other? In fact, whenever energy is transmitted from one body to another in time, there must be a medium or substance in which the energy exists after it leaves one body and before it reaches the other, for energy, as Torricelli[14] remarked, 'is a quintessence of so subtile a nature that it cannot be contained in any vessel except the inmost substance of material things' . Hence all these theories lead to the conception of a medium in which the propagation takes place, and if we admit this medium as an hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to endeavour to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise.

  1. Werke (Gottingen edition, 1867), vol.v. p.616.
  2. Abh. Leibnizens Ges., Leipzig (1846).
  3. Pogg. Ann., lxxiii. p. 229 (1848).
  4. Crelle s Journal, 72 (1870).
  5. Elektr. Maasb. inbesondere iiber das Princip der Erhaltung der Energie.
  6. Berlin Monatslericht, April 1872; Phil. Mag., Dec. 1872, Supp.
  7. In the whole of this investigation Weber adopts the electrodynamic system of units. In this treatise we always use the electromagnetic system. The electro-magnetic unit of current is to the electrodynamic unit in the ratio of to 1. Art. 526.
  8. March 19, 1845, Werke, bd. v. 629.
  9. Pogg., bd. cxxxv. 612.
  10. Tubingen, 1868.
  11. Mathematische Annalen, i. 317.
  12. Nuovo Cimento, xxvii (1868).
  13. Preface to Newton s Principia, 2nd edition.
  14. Lezioni Accademiche (Firenze, 1715), p. 25.