4.1 Heaviside, Poynting, & Energy Flow

"When Energy Goes From Place to Place, It Traverses the Intermediate Space"

Between them, Poynting and Heaviside solved the problem of following how electromagnetic energy moves from place to place. As Poynting noted:

If we believe in the continuity of the motion of energy, that is, if we believe that when it disappears at one point and reappears at another it must have passed through the intervening space, we are forced to conclude that the surrounding medium contains at least a part of the energy, and that it is capable of transferring it from point to point [[i]].

Similarly, Heaviside argued:

But if we can localise energy definitely in space, then we are bound to ask how energy gets from place to place. If it possessed continuity in time only, it might go out of existence at one place and come into existence simultaneously at another. This is sufficient for its conservation. This view, however, does not recommend itself. The alternative is to assert continuity of existence in space also, and to enunciate the principle thus:

When energy goes from place to place, it traverses the intermediate space [[ii]].

Figure 4.3: John Henry Poynting (1852–1914) [[iii]].

This is a foundational principle of electromagnetic theory. As Maxwell explained:

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 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 a hypothesis, I think it ought to occupy a prominent place in our investigations, and that we ought to endeavor to construct a mental representation of all the details of its action, and this has been my constant aim in this treatise [[iv]].

Maxwell’s contemporaries compared electricity to plumbing. “It is much easier to understand electricity when we assume it to be a fluid than in any other way, and as the fluid theory answers every purpose, we will assume it to be such,” explained the inaugural issue of The Electrician in 1882, “and the machinery for producing a current merely a pump for pumping or forcing electricity through wires and carbons as water may be forced through a pipe or valve” [[v]]. Figure 4.4(a) shows the fluid model for current flowing through a wire.

Figure 4.4(a) Traditional thinking interpreted electricity as a fluid contained and flowing within a pipe. (b) Heaviside and Poynting argued energy flows outside wires, conveyed by electric and magnetic fields at right angles to each other. (c) A planar cross-section view using dot-ex notation and showing the right-hand rule for the magnetic field.

Poynting and Heaviside argued that the plumbing and fluid analogy was completely wrong. Although current flows in the wire, the energy is carried by the fields around the wire, as in the three-dimensional representation of Figure 4.4(b) and the planar cross section of Figure 4.4(c). A few definitions provide common ground to consider the opposing models and sort out the physics.

Current (I) is the rate of flow of electric charge through a wire: how many Coulombs of charge flow in a second? One Coulomb per second is an Ampère of current. Voltage (V) is energy per charge. The rate at which the wire delivers energy, or power (P = I V), is the product of current and voltage. That much is the same in both theories.

In the fluid model, a battery or voltage source is like a pump, forcing an electric current through wires like a fluid flowing through a pipe. In analogy to the hydraulics of an incompressible fluid, the pressure is proportional to the energy per mass, and the electric fluid is capable of exerting an “electromotive force.” A resistor (R) is analogous to a turbine, converting the flow into heat energy. Wires convey energy due to the pressure and force of the electrical fluid within. This fluid model explained not only Ohm’s Law, but also the transfer of energy through and within the wires from the battery to the resistor.

In 1891, less than ten years after the inaugural issue that promoted the fluid theory of electricity, The Electrician hosted a lengthy running debate between John Toby Sprague (1824–1906), champion of the fluid model, and Professor Silvanus P. Thompson (1851–1916), champion of the Poynting-Heaviside theory. The anonymous referee offered a revised opinion of the fluid theory in his closing statement on the debate:

“The electromagnetic theory is certainly not simple, but it has much to recommend it ; while the idea that energy is conveyed along wires really had its origin, not in any definite theory, but in the fact that the wire was the only tangible thing about an electric current, and the energy was assumed to be connected with it simply because there was nothing else with which it could be associated…. [[vi]]

Let’s compare and contrast the two opposing views by considering their model of a simple electrical circuit.

Figure 4.5 A circuit (left) was interpreted as the flow of an electric fluid in analogy to plumbing (right) [[vii]].

Figure 4.5 presents the fluid model for a simple circuit with a battery connected to a resistor. The picture looks different from the Poynting-Heaviside model of Figure 4.6. The current flowing in a loop generates a magnetic field around the circuit. Applying the right-hand rule, magnetic field lines go down inside the current loop of the circuit and come up outside. The electric field lines (dashed lines) start on the top, positive-charged part of the circuit and end on the bottom, negative-charged part of the circuit.

Figure 4.6 The Poynting-Heaviside theory asserts energy (thick black lines) flows outside, not inside the wires. Power (P) may be obtained by multiplying voltage (V) and current (I) together. Demonstrating the equivalence of the Poynting-Heaviside theory requires the application of vector calculus to perform a complicated integration of the Poynting vector on the surface of the resistor.

In the Poynting-Heaviside model of Figure 4.6, the energy flows at right angles to the electric and magnetic fields–the dark black arrows connecting the source of the energy, the battery, to the load where the circuit delivers the energy, the resistor. Current flows inside the wires, but energy flows outside the wires, throughout the surrounding spaces around the circuit.

One can appreciate the skepticism directed at Poynting, Heaviside, and the advocates of the Maxwellian perspective. For more than fifty years, electricians used the intuition provided by the fluid-flow model, multiplying voltage and current together to obtain the power (P = V I) dissipated in a resistor. They had successfully applied the formula time and again in all manner of electrical applications, and it worked well particularly for Direct Current (DC) circuits and other where voltage and current only slowly changed.

Then, along came an unemployed telegrapher and a professor to claim their model was wrong. The energy flows around the circuit, outside, not inside the wires. An obscure and complicated vector calculus integration is required to demonstrate the mathematical equivalence to a result obtained far more easily the traditional way by simple multiplication. As Heaviside put it:

For without the mathematics and with only the sure knowledge of Ohm’s law and the old fashioned notions concerning the function of a conducting wire to guide one, no one would think of such a theory. It is quite preposterous from this point of view. Nevertheless, it is true; and the view was not put forward as a hypothesis, but as a plain matter of fact [[viii]].

The 1891 debate in the pages of The Electrician left the advocates of the electromagnetic theory triumphant, but the defenders of the fluid theory unpersuaded. The anonymous referee declared:

“…the idea that energy is located at all , and that, when it changes its position , it must move along a definite path, is quite a new one. The law of the conservation of energy implies that energy cannot disappear from one place without appearing in equal quantity somewhere else ; but, although this fact has been long accepted, it is only within the last few years that the idea of the transference of energy has been developed, or that anyone has attempted to trace out the actual path along which energy flows when it moves from place to place. The idea of an energy current is of more recent date than the electro-magnetic theory, and is not to be found explicitly stated anywhere in MAXWELL's work. We believe that the first time it was applied to electrical theory was in the pages of The Electrician, by Mr. OLIVER HEAVISIDE, to whom so much of the extension of MAXWELL's theory is due. The idea was also independently developed and brought to the notice of the Royal Society in a Paper by Prof. POYNTING….

“Until the electro- magnetic theory is further advanced it is unlikely that a discussion, such as Mr. SPRAGUE and Prof. THOMPSON have carried on, will alter the views of anyone already strongly biased upon the points at issue…. The theory explained by Prof. THOMPSON, however complex it may be, accounts exactly for the phenomena referred to, while the apparent simplicity of Mr. SPRAGUE's views is mainly due to their purely descriptive nature. …it is clear that further discussion can serve no useful purpose” [[ix]].

While Sprague had to acknowledge that energy could not be confined to wires in the face of Thompson’s examples of alternating current circuits, like the inductive coupling circuit of Fig. 4.7, he “decline[d] to accept Prof Thompson's assertion that ‘the energy paths in the medium are after all just as real entities as the magnetic lines are though not so familiar to us.’ No one has the right to make such a statement without some evidence” [[x]].

Figure 4.7 When switch K is closed, battery S launches a current through loop A, inducing a counter current in loop B, which may be measured by galvanometer G [[xii]].

Sprague concluded:

And all this structure of hypothesis and assertion is based upon the imaginary properties of an ether of which our only real know ledge is the single fact that light traverses space and our minds cannot conceive how it does so unless there is something there to transmit it [[xi]].

Next time: 4.2 Early Models of Electromagnetism: Ordered Abode of Reason or a Factory?

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References

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[i]  John Henry Poynting, "On the Transfer of Energy in the Electromagnetic Field," Philosophical Transactions of the Royal Instituion of Great Britain, 175 Part II, 1885, pp. 334-361.

[[ii]]        Heaviside, Oliver, Electromagnetic Theory, Vol. 1, London: Macmillan and Co. 1892, p. 73-74.

[[iii]]       See: https://en.wikipedia.org/wiki/John_Henry_Poynting#/media/File:John_Henry_Poynting.jpg.

[iv]  Maxwell’s Treatise vol. 2 p. 493.

[v] Hospitallier, E., “How the Current is Produced,” The Electrician, vol. 1, no. 1, 1882, p. 110. See: https://books.google.com/books?id=uYU5AQAAMAAJ&newbks=1&newbks_redir=0&dq=current%20fluid%20wire%20pipe&pg=PA110#v=onepage&q=current%20fluid%20wire%20pipe&f=false

[vi] Anonymous, “The Transfer of Energy,” The Electrician, vol. XXVII, no. 686, Friday July 10, 1891, pp. 270-272. See:
https://www.google.com/books/edition/The_Electrician/jxJbAAAAYAAJ

[[vii]]       Schantz, Hans Gregory, The Energy Flow and Frequency Spectrum About Electric and Magnetic Dipoles, Dissertation, University of Texas at Austin, August 1995, p. 27.

[viii] Heaviside, EMT vol. 1, p. 15.

[ix] Anonymous, “The Transfer of Energy,” The Electrician, vol. XXVII, no. 686, Friday July 10, 1891, pp. 270-272. See:
https://www.google.com/books/edition/The_Electrician/jxJbAAAAYAAJ

[x] Sprague, John T., “Electric and Magnetic Theories,” The Electrician, vol. XXVII, no. 686, Friday July 10, 1891, pp. 273-274. See:
https://www.google.com/books/edition/The_Electrician/jxJbAAAAYAAJ

[xi] Sprague, John T., “Electric and Magnetic Theories,” The Electrician, vol. XXVII, no. 686, Friday July 10, 1891, p. 274. See:
https://www.google.com/books/edition/The_Electrician/jxJbAAAAYAAJ

[xii] Thompson, Silvanus, “Electric and Magnetic Theories,” The Electrician, May 15, 1891, p. 44. Fig. 1.