The Efficacy of Imaginary Experiments
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In these essays I have used purely imaginary experiments, those devised by others and those that I have devised, to deduce laws of physics from more primitive axioms or propositions. Using the axiomatic-deductive method of reasoning, like Rumpelstiltskin weaving straw into gold, we weave fantasy into knowledge. How can we do that? What makes it possible?
It certainly goes against the Scientific Method that we were taught in school, the empirical-inductive method that Francis Bacon described in his science-fiction story AThe New Atlantis@ (written in 1624 and published, unfinished, in 1627, the year after Bacon died). In AThe Advancement of Learning@ (1605) Bacon had said that AThis is the foundation of all, for we are not to imagine or suppose, but to discover, what nature does or may be made to do.@ If we want to describe Reality and do so accurately, then certainly we must look at Reality, either through natural observation or through the contrived observation of an experiment. We cannot rightly expect to fantasize our way to true knowledge of Reality.
And yet the imaginary experiments laid out in these essays show us doing precisely that. Consider one way of deducing the existence of the electric displacement term in Ampere= s law. Although I have heard the story that has it attributed to James Clerk Maxwell, I have never read any evidence that he actually conducted this imaginary experiment: indeed, on reading his paper AOn Physical Lines of Force@ I was impressed by the observation that he snuck the term into Ampere=s law simply by deducing the fact that we must add some term proportional to the time-variation of the electric displacement field to the actual electric current density in order to give a correct account of basic electric phenomena involving matter. Nonetheless this legend of a fantasy still works for us.
Before Maxwell we would have had Ampere= s law in its most compact form as
that is, if we consider a region of space with electric current flowing through it and generating a magnetic field, if we multiply a minuscule element of spatial displacement (dl) by the component of the magnetic induction field (B) parallel to the element at one of the points on that element (the vector dot product), and if we integrate that product for all elements of spatial displacement forming a closed loop (Ő ), then we will obtain a number equal to the product of the magnetic permeability of vacuum and the amount of electric current (I) penetrating the surface bounded by the loop. In the simplest possible example, an extremely long straight wire carries an electric current I and generates a magnetic field B(r) a distance r from the wire such that the integration yields
Imagine cutting a short segment out of that wire and brazing flat metal plates to the cut ends of the wire in such a way that they sit parallel to each other and perpendicular to the wire. The current in the wire will eventually stop flowing, but, because the electric charges accumulating on the plates attract each other across the gap between them, the current will flow longer or stronger than it would do without the plates. Also, the electric current will no longer flow steadily, but it will, nonetheless, flow and at every instant, whether the current flows or not, Ampere=s law must remain true to Reality.
We tacitly assume that the loop over which we integrate the magnetic induction field bounds a circular disc through whose center the wire passes. That assumption greatly simplifies the analysis of the problem and does not interfere with the generality of the result, so I will keep it here. However, I want to make the surface bounded by the loop so elastic that I can stretch it in any way that I need. Imagine so stretching that surface that it extends alongside the wire and then passes through the gap between the plates, nowhere so much as touching metal. We now seem to have contrived a violation of Ampere=s law as expressed in Equation 1: at any instant when an electric current flows in the wire a magnetic field exists on the loop going around the wire (giving the integral on the left side of Equation 1 a non-zero value) but no electric current penetrates the surface bounded by the loop (giving the right side of Equation 1 a value of zero). True, Equation exists as a mere description of a part of Reality, but the notion that we can introduce inconsistencies into that description by moving a purely imaginary surface offers us a challenge: How accurate is our description of Reality?
In this case we meet the challenge by finding something in our imaginary experiment that correlates with the current in the wire. Electric current measures the rate at which electric charge passes a point, so electric current correlates with the rate at which electric charge builds up on or drains out of the plates. That change in the amount of electric charge on the plates, in its turn, makes the strength of the electric field in the gap between the plates increase or decrease. That electric field penetrates the stretched surface, so it must, in some way, correspond to the missing electric current in Ampere=s law.
We can represent the stretched surface as a crazy quilt of minuscule rhombi, most of them square (or at least approximately so). Each rhombus has an area (da) that we represent as a vector with an arrow perpendicular to the rhombus and rising outward (relative to the whole surface) from its center. We multiply that area vector by the parallel component of the vector representing the rate at which the electric displacement field changes (ε0 M E/M t). Integrating that product over the entire surface (S) gives us a mathematical representation of the rate at which the flux of the electric displacement changes. Maxwell called that the electric displacement current and added it indirectly to Ampere=s law (by adding it to the current of what he called Afree electricity@ ) to get
Does a time-varying electric field actually generate a magnetic field, as that equation implies? No experiment involving the examination of electric currents in themselves can answer that question unambiguously. Maxwell took an indirect approach. He combined that equation with Faraday=s law of electromagnetic induction and devised a theory describing the existence and properties of electromagnetic waves. If someone could prove and verify that theory, then they would also verify the truth to Reality of the proposition added to Ampere=s law. In 1886 Heinrich Hertz began conducting a series of experiments in which he did just that (and also discovered the photoelectric effect, which led eventually to the quantum theory of light).
So, as Maxwell did, we got it right. But what makes that possible? Maxwell didn=t create the law by imagining it: vibrating electric fields have been generating magnetic fields since the Universe began its existence. How could Maxwell, with only his imagination, discover an unknown law of physics?
Electricity and magnetism do not give us directly the percepts from which we can conceive our ideas of them, as we do with objects such as amber, copper wire, etc. We obtain our knowledge of those two phenomena from the effects they impose upon things from which we do get percepts. Thus we stand doubly removed from the thing-in-itself in this study. That separation notwithstanding, in the two centuries that elapsed between the publication of William Gilbert=s Ade magnete@ (1600) and Allesandro Volta=s invention of the electric pile (1801) natural philosophers discovered enough knowledge about electricity and magnetism that the physicists of the Nineteenth Century could perform the advanced experiments required to support the creation of a complete electromagnetic theory.
People have known since antiquity that if they rub fur on glass or amber (Greek elektron, from which we derive the word electricity), then small, light things, such as chaff or bits of paper, would move when the rubbed objects came near them. But the rubbed objects emit no percepts different from the ones they emit before they are rubbed together. Whatever makes the chaff and the bits of paper move is thus insensible: we can only infer its existence and explore its properties through its effects on small, light things.
That=s just what the scientists of the Seventeenth and Eighteenth Centuries did. They observed what they could of all known electric phenomena, then they devised experiments to contrive further observations. At first they used bits of paper as electrical detectors, but then they discovered that two leaves of gold foil suspended in a glass jar worked even better: when an electrified object touched the brass knob on top of the jar=s lid the gold leaves would spread away from each other. Benjamin Franklin used such a detector to show him that a thunderstorm into which he flew a kite had electrified the kite= s string.
From the percepts emanating from such detectors scientists were able to conceive and refine their basic ideas of electricity and then to confirm that those ideas mimic the behavior of electricity-in-itself. They thus established the basic facts of electricity:
1. electricity occurs in two kinds (called charges) B vitreous (now called positive) and resinous (now called negative) B in equal amounts;
2. like charges repel each other and opposite charges attract each other in accordance with an inverse-square law (Coulomb=s law) similar to Newton= s law of gravity;
3. electrification occurs by the removal of one kind of electricity from one object and the deposit of it onto another object;
4. electricity moves readily through some materials (conductors, usually metals) and not at all or very slowly through others (insulators, usually ceramics); and
5. electricity moves as a fluid, in currents, and great electrical currents flow within thunderstorms, especially in lightning.
In 1801 Alessandro Volta, after repeating experiments that Luigi Galvani had conducted with electricity and frogs= legs, parlayed the knowledge that he had thus gained into his invention of the electric pile, an alternating sequence of copper and zinc discs separated from each other by paper or cloth soaked in acid. By attaching a wire to the uppermost or lowermost disc in the pile and then touching its free end to various objects, especially the gold-leaf electricity detector, Volta obtained the percepts that indicated to him that the pile was electrifying those objects. From his electric pile Volta obtained vastly more electricity than he could have obtained from even the largest friction-based electrifiers of the time, so he inferred that the electric pile could produce relatively strong electric currents for significant intervals of time. That fact enabled the physicists of the Nineteenth Century to conduct the more demanding experiments that would reveal features of electricity and magnetism that remained invisible to previous explorers of this realm, features that would lead to a complete and fully traversable electromagnetic theory.
Hans Christian ěrsted performed the first experiment, in the summer of 1820, to demonstrate a connection between electricity and magnetism. As with electricity, people had known since antiquity that certain minerals (lodestones) can affect iron at a distance. They observed that a lodestone can alter the pointing of a navigator=s compass and inferred from that percept the idea that the compass detects a magnetic force (which led William Gilbert, in 1600, to propose in his book de Magnete the idea that Earth is a giant lodestone). Travelers, using magnetic compasses to find their way through Europe=s forests and mountains, noticed that the magnetized needles changed their orientations erratically during thunderstorms. Correlation of the percepts (those of the thunderstorm and of the erratic turnings of the needles) led ěrsted to conceive the idea of a thunderstorm as a magnetic phenomenon. Benjamin Franklin had already proven and verified the proposition that thunderstorms have an electric nature, so ěrsted sought to explore the connection, if any, between electricity and magnetism as implied by that twin nature of thunderstorms.
To put his hypothesis to the test, ěrsted suspended a thin straight wire over a table and connected it, through a switch, to a voltaic electric pile. Then he set a magnetic compass under the wire. When he closed the switch and thereby completed the electric circuit, everything he knew of electrical conductors and voltaic piles told him that electric current (he called it a conflict between the electricities) flowed through his wire. At the same time he saw the needle of his compass turn. When he opened the switch and thereby interrupted the flow of electric current, he saw the compass needle turn back to its original orientation. Repeating the experiment under different circumstances gave him a simple correlation between two percepts B the closing/opening of the switch and the turning of the magnetized needle. That association of percepts leads to an association of the related concepts B electric current with magnetic force B so that when we receive the percepts associated with the flow of an electric current we can expect, under the right circumstances, to receive the percepts of the exertion of a magnetic force.
The year after ěrsted conducted his experiment Andre-Marie Ampere enlarged the concept of electromagnetism by way of Newton= s third law of motion. Ampere reasoned that because an electric current exerts a magnetic force, it must also receive a magnetic force. With that concept in mind Ampere conceived, designed, and performed four experiments in which he used only wires carrying electric current to produce and to respond to magnetic forces. From the data that he obtained from those experiments Ampere extracted a mathematical description of the magnetic force that two electric currents exert each upon the other. His algebraic formula was a clumsy one, but over the next few decades other mathematical physicists transformed it into more convenient forms, such as the Biot-Savart law and the truly elegant Ampere= s law, Equation 1 and its differential equivalent (L H H=j).
Michael Faraday conducted his own versions of ěrsted=s experiment. He created various arrays of wires penetrating sheets of paper covered with iron filings and noticed that when electric current flowed through the wires the iron filings responded to the magnetic force by turning like tiny compass needles (which he expected) and formed arrays reminiscent of a plowed field (which he didn=t expect). That similarity inspired him to conceive the idea of a forcefield (purely as an Aaid to the imagination@, you understand), which concept corresponds to the statement that an electric current fills the space around it with a condition that, at each point, exerts a magnetic force upon any magnetic charge or electric current at that point. Here again we see a percept transformed into a concept and given a name in accordance with a similar-looking percept. Also by analogy we can conceive the idea of an electric forcefield emanating from electric charges.
But do forcefields truly exist? By that question I mean to ask Can we properly include forcefields as elements of the set that we call Reality? Or must we dismiss them as mere figments of human imagination, ghosts of a refusal to accept spooky actions at a distance? Note that we don=t get percepts from fields directly, as from bodies, but indirectly through their effects on bodies. Can we truly attest the existence of a phenomenon that we can only apprehend indirectly? At first that question got a rousing No! as its answer.
At the beginning of modern physics in the Seventeenth Century natural philosophers banished the idea of action at a distance to the realm of spooks and ghosts, much as Einstein later rejected the idea of quantum entanglement as Aspooky actions at a distance@ . It was the horror of that concept (and of the apparently intractable problem it represents) that moved Isaac Newton to proclaim AI frame no hypotheses@ with regard to the cause of gravity. Earlier Galileo had criticized Johannes Kepler for having Alent his ear and given his assent to the moon=s dominion over the waters, to occult properties and to such puerilities.@ Actually Kepler was guilty of proposing something like the modern concept of gravity, of like attracting like offering an explanation of how the sun holds the planets in their orbits and how the moon raises the tides.
Given that people of that time were just getting accustomed to reasoning from the evidence of their senses, rather than from the Awisdom of the Ancients@ , that attitude seems understandable. But now our senses bring us evidences of things that we cannot perceive directly and we must decide what to make of that fact. Can we actually accept phantasms as elements of Reality?
Even before Maxwell conceived his imaginary experiment (and there seems precious little evidence that he actually made his discovery that way), the theory of electromagnetism, such as it was, had intruded into the realm of fantasy. Recall that Ampere=s law has us calculating a line integral of the magnetic field around a closed loop and correlating that with the amount of electric current flowing through a surface bounded by the loop. But neither the loop nor the surface actually exist: they fall entirely into the class of entities that Faraday called Aaids to the imagination@. We merely imagine these structures marked out in space and use them to guide our calculations. And yet we find that Reality has such a form that those calculations give us a valid law of physics, even though the first part of the calculation does not give us a description of anything we can observer or measure (as, for example, Newton=s law of gravity allows us to calculate the rate at which a body would fall toward a planet).
So, according to the story, Maxwell took into his mind the knowledge that those others had acquired and conceived his own version of ěrsted=s experiment. In the laboratory of his imagination he envisioned having a voltaic pile connected to a loop of wire through a switch. The images that he conceived mimicked percepts that he had received from real electric batteries and wires, so he was justified in assuming into his reasoning the proposition that when he imagined closing the switch, electric current would flow in the wire and that a magnetic field would come into being around the wire. The magnetic forcefield, of course, mimicked the entity whose existence Faraday had inferred from his iron-filing studies. At that point Maxwell had not yet done anything in his imagination that others had not done in the real world.
Next he conceived the construction of a circle whose axis coincided with a long straight segment of the wire and a disc bound by that circle. Like the lines of latitude and longitude drawn on a map, these constructions are purely imaginary entities that we conceive solely for the sake of supporting human calculation. In this case the calculations consist of a closed line integral of the magnetic field on the circle and a surface integral of the electric current density penetrating the disc. Ampere= s law relates those calculations to each other through the magnetic permeability of vacuum, a number that converts units of squared electric current divided by squared distance into units of force.
With imaginary wire cutters Maxwell clipped a short segment out of his wire some distance from the point at the center of his imaginary circle and attached flat metal plates to the cut ends of the wire. He knew that the electric current would still flow in the wire as electric charge built up on the plates, positive on one plate and negative on the other. In that concept he mimicked one of the experiments that Alessandro Volta had performed to prove and to verify the proposition that his electric pile did, indeed, electrify objects by pushing electric current into them under high electric pressure (which we now call voltage in his honor).
Finally Maxwell took the step that no one had taken before. He moved his imaginary disc into the space between the plates and extended an imaginary cylinder from its rim to the circle on which he integrated the magnetic field, thereby keeping the circle as the disc=s boundary. At any instant when electric current flowed in the wire, then, his loop integral in Ampere= s law would still yield a number but the corresponding surface integral would yield no corresponding number because no electric current penetrated the disc. That fact gave Maxwell two possible ways to save Ampere=s law: 1. either he could somehow restrict the geometric figures over which we can carry out the integrations of Ampere= s law or 2. he could add some missing term to Ampere=s law.
Can moving a purely imaginary surface alter one of the calculations in Ampere=s law? If we say yes, then we have two possibilities in consequence; either the change in the surface integral makes the loop integral change or Ampere=s law does not correctly describe Reality in all circumstances. We dismiss the first possibility immediately: it would require a real object (the magnetic field) to change in response to the movement of a purely imaginary object (the surface of integration). Does that then mean that the validity of Ampere=s law depends upon where we carry out the relevant integrations? Maxwell would have said no, of course not, and dismissed Possibility #1 altogether. Before we do that let=s take a sharper look at that last question.
This discussion will take us deep into the relationship between mathematics and Reality, so I have set it off as a separate section in case you want to scroll ahead to the rest of my analysis of Maxwell=s imaginary experiment.
The mathematics that we use in the physical sciences constitutes an axiomatic-deductive structure. As I have shown in essays elsewhere on this website, we can begin with an ordered list of names (the counting numbers) and deduce the existence of new kinds of numbers and methods of combining numbers in ways that we can use to describe Reality. Of course we begin by describing things that we actually perceive and measure.
Galileo Galilei offers an excellent example in his analysis of balls rolling down inclined planes. Historian Stillman Drake has offered a nice hypothesis based on the observation that Galileo came from a family of professional musicians. According to Drake=s hypothesis, Galileo fitted his grooved planks with moveable frets so that the balls would make little bumping sounds when they rolled over them. Modern research has confirmed the fact that musicians can accurately divide time into very short equally long intervals, so Drake supposes that Galileo rolled balls down his planks and then adjusted the spacing of the frets until the bumping of the balls kept perfect time with a brisk little song that he sang. Galileo then knew that the spacing of the frets on the plank represented equal intervals of time.
Galileo then used barleycorns to measure the distances between the frets. He discovered that the ratios of the distances to the length of the first interval came close enough to the sequence of the odd integers that he could proclaim equality. As a mathematician, he knew that the sum of the first N odd integers equals the square of N, so he inferred that the distance that a uniformly accelerating object travels is proportional to the square of the elapsed time with the proportionality factor equal to half of the acceleration. Thus Galileo began the mathematization of physics.
Using the calculus developed by Isaac Newton and Gottfried Leibnitz, we can describe the motion of a body accelerating at an unvarying rate A. During any minuscule interval of time dt the body gains a minuscule amount of speed dV=Adt. We add those increments together over the interval t and find that the body gains the speed V=At from the time that it was released. At any given instant that we mark with the elapsed time the body crosses a distance dS=Vdt=Atdt. We add up the increments of the accruing distances through the process of integration to find out that over the interval t since it was released the body travels a distance
Thus by purely conceptual means, beginning with defining velocity as the ratio of distance crossed to time elapsed and acceleration as the temporal rate at which a body=s velocity changes, we have deduced, by way of the calculus, a description of a body=s motion that precisely matches the description that Galileo obtained by applying induction to the data that he obtained from his experiments. By juxtaposing the percepts from his rhythmic singing and a handful of barley with those he received from his rolling balls and fretted planks Galileo put numbers on distance and duration and then correlated them through a simple algebraic formula. Physical Reality seems to mimic the mathematics that we conceive as an exercise in pure logic.
Now consider one way in which Isaac Newton could have devised his law of gravity and compare it to the way he actually did it (in Book III of the Principia). The way he actually did it is too complicated for this brief digression.
Once he had dismissed, through his first law of motion, the idea that celestial bodies possess a natural propensity to move in circles, Isaac Newton then had to work out a clear description of the acceleration that a body moving at speed V has to suffer toward the center of a circle of radius R in order to remain on the circle. Although he may have used sketched diagrams as aids to the imagination (actually, more as aids to memory so that he wouldn=t have to struggle to keep everything clear in his mind while he was focusing on one part of it), that derivation was purely conceptual. Sure, we could devise an experiment that would give us the perceptual data from which we could obtain a description of centripetal acceleration by induction, but we know that we don= t have to: we can use purely conceptual deduction instead.
Imagine a circle of radius R with a body moving on it at a uniform speed V. Draw a radius vector, a straight line extending from the center of the circle to a point on the circle itself, and then draw a second radius vector separated from the first by a minuscule angle so small that the length of the arc spanning the gap between the radius vectors= heads differs negligibly from the length of the arc= s chord. That chord represents the distance the body moves in a minuscule interval of time dt (dS=Vdt). It also coincides with the base of an isoceles triangle whose sides coincide with the radius vectors.
From the head of each radius vector draw a straight line tangent to the circle at that point. Give both of those lines the same length and identify them as representing the velocity the body has when it occupies the point on the circle indicated by the radius vector whence the line emanates. Without changing the orientation of those velocity vectors, slide their tails along the arc between the radius vectors until they coincide and then draw a straight line from the head of one velocity vector to the head of the other. That line represents the change that occurred in the body=s velocity in the interval dt; that is, dV=Adt, in which A represents the body=s centripetal acceleration.
We now have two isoceles triangles. Because we drew the velocity vectors perpendicular to their respective radius vectors, we know that both triangles have the same vertex angles between their sides, so we have two similar isoceles triangles. That fact necessitates that we have the same ratio between the base and one side for both of them; that is,
which leads immediately to
Though you might have used a sketch as an aid to memory, we nonetheless count this as a purely conceptual derivation of a description of the motion of a body moving on a circle. Now consider some bodies that actually move on paths closely resembling circles B the planets. From percepts obtained through their measuring instruments, with the aid of telescopes after about 1610, astronomers had, by 1687, abstracted descriptions of the planets= orbits, fairly accurate ones so long as they were expressed as ratios relative to Earth=s orbit. So in the table below I list the names of the planets in Column 1 [note that Newton did not know about Uranus and Neptune and, yes, I still regard Pluto as a planet (I learned my basic astronomy long before the first Kuiper Belt objects were discovered in the late 1970's): I have left Pluto off the table because its orbit is just too elliptical to approximate as a circle.]. In Column 2 I list the mean radii of the planets= orbits in Astronomical Units. In Column 3 I list the planets= orbital periods in years. In Column 4 I list the results of dividing the entry in Column 2 by the entry in Column 3, giving a measure of the planet= s orbital speed (the units, if we want to present these ratios as velocities, would be astronomical units per year and they would be smaller than the actual orbital speeds by a factor of two pi). In Column 5 I list the result of squaring that quasi-velocity and then dividing by the orbit= s radius to obtain a measure of the centripetal acceleration needed to keep the planet on the orbit. In Column 6 I list the squares of the orbital radii and in Column 7 I list the result of multiplying those by the entries in Column 5 (note that most of those products differ from a precise 1.0000... because I have used rounded off numbers and ignored the fact that the orbits are slightly elliptical):
Clearly the phenomenon that keeps the planets in their orbits, if it emanates from the sun, conforms to a description that makes the imposed centripetal acceleration equal to some constant (the sun=s mass multiplied by the Newtonian gravitational constant) divided by the square of the distance between the sun=s center and the orbit. Thus Newton was able to take a purely conceptual description of motion, obtained from his strange dynamic geometry, correlate it with numbers obtained from observations of certain features of Reality, and devise his law of gravity. As to what causes this phenomenon of gravitation, he couldn=t say and he refused to speculate (AI frame no hypotheses,@ he said).
Charles Augustin Coulomb discovered his own inverse-square law, the one pertaining to electric force, using a torsion balance, which he invented, and revealed it in 1788. By Coulomb=s law a particle carrying an electric charge q exerts upon a particle carrying an electric charge Q at a distance R from the first particle a force equal to
in which equationε0 represents the electric permittivity of vacuum. The factor 1/4πε0 provides the electric analogue of Newton=s gravitational constant. Mathematical physicists of the Nineteenth Century put it into that particular form because of something attributed to Karl Friedrich Gauss.
After Michael Faraday conceived the idea of a magnetic forcefield it didn=t take long for someone to conceive the idea of an electric forcefield. If we divide Equation 7 by q, we get
which describes a potential force waiting at every point a distance R from the particle carrying electric charge Q for some particle carrying electric charge q to come and actualize that force. Every electric charge thus envelopes itself in a kind of Štherial cloud. Physicists in essence conceived the electric forcefield as an abstraction derived from Coulomb=s percepts pertaining to the forces exerted between two electrified bodies.
What Gauss did should properly astound us. He started by defining electric flux as the product of the electric field strength at some point and the area of a small patch of surface on which that point lies. More precisely, on a minuscule patch of surface the flux equals the product of the patch=s area (dS) and that part of the electric field that points in the direction perpendicular to all straight lines tangent to the patch (Ez). Gauss imagined a closed surface as a patchwork quilt of such elements, calculated the total electric flux through that surface by the process of integration, and discovered that the total electric flux through any closed surface equals the net electric charge enclosed within that surface divided by the electric permittivity of space;
In that equation we represent electric charge density by the letter rho and note that the circle on the integration sign indicates that the surface S is a closed surface, the boundary of the volume V. We call that equation Gauss=s law for the electric field and count it as the first of Maxwell=s Equations.
Thus we see that Existence has so shaped Reality that we can use purely imaginary geometric forms and their boundaries to correlate our mathematical descriptions of the phenomena of electricity and magnetism. We might call a generalized version of that statement Gauss=s law of Rationalism. We can certainly regard it as a second level of abstraction over the basic laws of physics. In Coulomb=s law we have a concept abstracted from percepts, one that we can manipulate to devise electric situations that will yield new percepts that will match our expectations. In Gauss=s law we have what appears to us as nothing more than a fancy device for bypassing Coulombic calculations in devising descriptions of electric fields. Electric flux otherwise has no meaning: it doesn=t correlate with any percepts. Nonetheless, when we apply that law, insofar as we can do so, we find that it matches our perception of Reality.
Ampere=s law gives us another example of this phenomenon. Instead of a volume bounded by a closed surface, we use an open surface bounded by a closed line. In this case we have the magnetic field integrated around the closed line correlated with the amount of electric current flowing through the surface bounded by the line. Again we have a source in a geometric figure correlated with its forcefield on the boundary of that figure. But unlike the case with Gauss=s law, in which the boundary encloses a unique volume, the closed line in Ampere=s law bounds an infinite set of surfaces. Which of those surfaces belong in Ampere=s law and which do not?
Because geometric figures exist only and entirely in our imaginations, they can have no effect upon the laws of physics. We can only use them in describing those laws. So what choice of surfaces merely describes Ampere=s law and does not prescribe it? Maxwell=s imaginary experiment offers us a means of answering that question.
Maxwell knew that stretching the surface of integration and slipping it between the plates in his imaginary apparatus would not change the integration of the magnetic field on the surface= s boundary. But the electric current does not penetrate that surface, so Maxwell had to find some other feature of the experiment to integrate on the surface to maintain the validity of the loop integral. That something would have to coordinate somehow with the electric current. The electric current coordinated with electric charge building up on or draining out of the metal plate between the surface and the loop of integration and that coordinated with an electric field in the gap between the two plates strengthening or weakening. And that electric field penetrated Maxwell=s surface, so he could integrate it on the surface as a changing electric flux and then multiply the result by the electric permittivity of vacuum to obtain what Maxwell called Athe general displacement of the electricity@. Maxwell added that to Ampere=s law to get
in which i represents the electric current density (Maxwell called it the electric conduction current to distinguish it from the electric displacement current) and we have the electric displacement field, D=ε0E.
Maxwell didn=t extract that addition from any percepts that emanate from electric displacement. He noticed that of all the surfaces that could go into Ampere=s law a small subset, those that went through the gap between his metal plates, seemed to have an exemption from participation in that law. He chose to believe that Reality does not grant exemptions from its laws to purely imaginary constructs and Reality seems to agree.
Combining his version of Ampere=s law with Faraday=s law of electromagnetic induction, Maxwell derived an equation that describes mutually-supporting electric and magnetic fields propagating through space. From the constants in the equation, the electric permittivity and the magnetic permeability of vacuum, he calculated the propagation speed of electric waves and found that it came close enough to the measured speed of light that he could assert an equality and add to his hypothesis a statement that light consists of very high frequency electromagnetic waves.
Beginning in 1887 Heinrich Hertz obtained the proof and verification that science requires to transform Maxwell=s hypothesis into theory. Hertz built an apparatus that would give him percepts that he could only interpret through Maxwell=s hypothesis. The first part of the apparatus, on one side of Hertz=s laboratory, consisted of a high-voltage circuit that would make a spark jump across a gap located at the focus of a parabolic reflector. The second part, on the opposite side of the laboratory, consisted of a loop of wire with a narrow gap located at the focus of a second parabolic reflector set up to face the first one. Hertz saw that when he made an electric spark jump the gap in the first part of the apparatus a spark also jumped the gap in the second part, even though that part of his apparatus was not connected to any source of electricity. According to Maxwell=s hypothesis, the first spark constituted a rapidly varying electric current that generated an electromagnetic wave that spread outward in all directions. Part of that wave struck the first parabolic reflector, which reshaped it into a collimated beam aimed at the second reflector, which reshaped it into a cylindrical wave converging on the gap in the wire loop, where it created an electric field strong enough to force a spark to jump the gap. Light emanating from the sparks gave Hertz the percepts that validated Maxwell=s concepts, both the wave equation and the addition to Ampere=s law that produced it.
Now we know that at two levels removed from direct copying of percepts, human concepts derived through logic faithfully mimic the structure of Reality. We can conceive a kind of ghostliness to this business. We have deduced a description of Reality that is correct (insofar as we can test it) and yet our description seems to stand aloof from what it describes. We talk about measuring distances and durations with measuring rods and clocks, but we never actually grasp the fabric of space and time. We convert percepts into concepts and then use logic to transform those concepts into yet other concepts that guide us in devising phenomena whose percepts actually match what our concepts led us to expect.
How Maxwell Actually Did It
I first heard the story of Maxwell=s discovery of the electric displacement term in Ampere=s law in the late 1960's when I was studying physics at UCLA. It seems entirely plausible and I have no evidence to indicate that Maxwell did not, at some point, conceive such an idea. But looking at Maxwell=s major writings offers a different view of what likely happened. The derivation that I described above remains valid: the question is whether Maxwell did it.
In Maxwell=s paper AOn Faraday= s Lines of Force@ (1856) we see a hint that Maxwell may have had our imaginary experiment in mind at the time. Near the end of the paper, in the section he titled ASummary of the Theory of the Electro-tonic State@, he states Law III (of four) as, AThe entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface.@ That statement gives us a very good verbal description of Equation 1, Ampere=s law in its primitive form. The reference to Aany surface@ hints at the imaginary experiment.
But in his paper AOn Physical Lines of Force@ (1861) Maxwell did something completely different. In the first paper he gave a mostly verbal description of Faraday=s concept of a forcefield and of the electro-tonic state (what we now call the magnetic vector potential), while in the second paper he went deeply into abstract mathematical descriptions of the phenomena (at which he excelled) based upon a hypothesis of the fields as expressions of Štherial vortices that he presumed to fill space. In that latter paper Maxwell added the electric displacement term to Ampere=s law almost casually. Near the end of the paper he defines Athe general displacement of the electricity@:
AIn a dielectric under induction, we may conceive that the electricity in each molecule is so displaced that one side is rendered positively, and the other negatively electrical, but that the electricity remains entirely connected with the molecule, and does not pass from one molecule to another.
AThe effect of this action on the whole dielectric mass is to produce a general displacement of the electricity in a certain direction. This displacement does not amount to a current, because when it has attained a certain value it remains constant, but it is the commencement of a current, and its variations constitute currents in the positive or negative direction, according as the displacement is increasing or diminishing. The amount of the displacement depends on the nature of the body, and on the electromotive force; so that if h is the displacement, R the electromotive force, and E a coefficient depending on the nature of the dielectric,
and if r is the value of the electric current due to the displacement,
So Maxwell conceived the polarization of a dielectric medium by an applied electric field as a displacement of electric charge that produced its own field in response. He then identified any change in that displacement with an electric current, the electric displacement current. Five pages further on Maxwell comes to Proposition XIV: ATo correct the equations (9) of electric currents for the effect due to the elasticity of the medium.@ Though he doesn=t mention Ampere, the equations to which he refers are the differential-equations version of Ampere=s law, although his archaic mathematical notation and the fact that he inferred the equations as part of another derivation tend to obscure that fact. And the elasticity to which he refers is an electric elasticity of the kind we find in dielectric media, the kind that permits electric displacement to occur. As he notes, A...a variation of displacement is equivalent to a current, and this current must be taken into account in equations (9)....@ So he expressed the displacement current in terms of the electromotive force that induced it and added that to Ampere=s law to get three equations;
in which p, q, and r represent the electric currents in the x, y, and z-directions respectively; alpha, beta, and gamma represent the components of the magnetic field; and P, Q, and R represent the electromotive forces associated with the electric displacement. In modern notation those equations become
With a little rearranging that gives us the fourth of Maxwell=s Equations, Maxwell=s version of Ampere=s law.
We can=t help but notice that Maxwell used differential equations rather than integral equations in his paper. That fact tends to argue against the story of the imaginary experiment, because that story depends very much on the use of integrals in its formulation. As a rule physicists prefer to use differential equations to express the laws of physics and to manipulate them. The integral versions of physical laws appeal more to engineers. Still, it=s a good story.
Tentative Axioms of The Map of Physics
If The Map of Physics is to constitute a complete axiomatic-deductive system, we must found it on a small set of axioms, as Euclid did with plane geometry. Here is a preliminary list of those axioms.
Existence consists of things participating in events within the Frame of Reality, a thing that has a unique and consistent existence independent of us.
Occam=s Razor. Entities are not to be multiplied beyond necessity. We thus seek to study the necessary/essential and to eliminate from our consideration the contingent/accidental.
The Principle of Sufficient Reason. If something is so which could have been otherwise, then there must be some reason why it is so, and not otherwise. It takes a difference to make a difference.
EPR (Albert Einstein, Boris Podolsky, Nathan Rosen); AIf, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.@
Combining Axiom I and Axiom II leads to Naturalism, which denotes the idea that we can explain the Universe and all of its phenomena entirely in terms of natural phenomena. We shall invoke no ghosts, spooks, or any other denizens of the spirit world.
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