Saturday, 17 July 2010
A new theory of Magnetic Fields
The tip of the iceberg
When man first discovered that a piece of loadstone hanging on a thread would always point in the same direction; when Oersted discovered that an electric current in wire affected a compass needle; when Faraday discovered that electricity can be generated by moving a magnet inside a wire coil; they perceived only the tip of the iceberg. The phenomena which we observe and have used to construct wonderful technologies are just the by products of the true nature of magnetism.
What I will show is that magnetism is as fundamental to the structure of matter as the electric force which binds the negative electrons to their positive nuclei. As we delve into the inner mechanisms of nature, magnetism becomes ever more significant. It is the regulator of processes. The phenomena we observe in the macro world are bye products of the inner workings of matter.
Without this insight, Oersted, Ampere, Faraday, Gauss, Biot, Savart and Michelson were working in the dark. The laws they developed are not fundamental laws of nature, but mathematical models designed to mimic the observed phenomena. As a result, they are not wholly self consistent and do not make complete sense. I remember sitting in a lecture trying to follow the mathematics. I am dyslexic and so was unable to take meaningful notes. The other students were writing everything down without understanding. I was more concerned with the way a term of seemed to come and go from equations. This was before the days of S.I. Units and there were four or five systems of units in operation. I plucked up courage and interrupted the lecture. It was easy with only 20 students in the cosy little physics department of Royal Holloway College.
"Excuse me Doctor F..... , but ......"
My memory is not good enough to recount the exact details, but it transpired under cross examination that the good doctor had compiled his notes from two text books. What he had not realised was that the two books were based on two different sets of units. In one set, an equation would need a term of , while in the other it would be absent. You will not be surprised by the good doctor's reply.
"I did that to keep you on your toes"
I tell this tale because it is symptomatic of the way in which knowledge is passed from one generation to the next. All too often, teachers do not understand what they are teaching. They have learnt it, not understood it. The students blindly copy down the notes and they in turn learn it without understanding.
Being dyslexic, I am a stranger to this process. It has taken me 40 years to acquire the technology and learn how to use it to overcome this disability. I could not record information in writing and recover it. My disability meant that any notes I made were not comprehensible to me. I never had the luxury of being able to learn. I had to understand. I have to understand. If I cannot understand something, I cannot capture it in my mind. If it has a structure which I can comprehend, its in for good.
In the enforced leisure of my early retirement, I have been revising my physics. I found that I could not understand the principles of electricity and magnetism. So I did what I should have done 30 years ago and worked out the theory for myself. Searching out and wrestling with the paradoxes, I hope I now comprehend the subject. In the film "The man went up a hill and came down a mountain" there are two Welsh farmers. Brothers, they are and know locally as Thomas Tupp and Thomas Tupp Too. The word tupp means "thick" or stupid, or educationally disadvantaged. One of the brothers introduces them.
"I'm Thomas Tupp and this is my brother Thomas Tupp Too.
"Folk say we are tupp, but we are not so tupp that we do not know that we are tupp."
And therein lies the dilemma of the scientific community. When people do not realise their own intellectual limits, they do not enter into the struggle to understand that which they have failed to comprehend. Learning is no substitute for understanding. It is in admitting a lack of understanding and in wrestling with the problem that the mistakes of the past are rectified. I came across a paper by TFG Searle, a colleague of JJ Thompson. He was looking at the magnetic field which should in theory surround a moving charge. This was just a few years after the discovery of the electron. JJ Thompson devoted a vast amount of energy to trying to explain the existence of matter in terms of vortices in the aether. Searle's paper tried to show that a moving spherical charge would posses the property of inertia. It does not manage to do that but ends in hope that he might be able to do so one day. A second paper on the subject nearly ten years latter is a load of rubbish.
This is the basic concept. A moving charge generates a magnetic field. Magnetic fields contain energy. To accelerate a charge, you must do work in order to create the magnetic field. So the charge resists being accelerated with a kind of inertial force. The concept is correct, but existing electromagnetic theory makes the task of calculating this impossible. The law of Biot-Savart tells us the magnetic field must be
When we accelerate the charge, increases. Magnetic flux moves outwards from the surface of the charge. Faraday's law says that this movement generates an electric field. This electric field can then act on the surface of the charge and produce a force. But there is a problem. When we try to calculate the number of lines of force which surround the charge, this is the integral of over a half plane extending to infinity from the line of motion of the charge, we get infinity. If Faraday is right, then nothing in the universe could move.
What I have done is to work out how magnetic flux must behave in order to overcome this paradox. It turns out that magnetism gives charges the property of inertia. We find that what we had called mass can be accounted for entirely in terms of energy in magnetic fields. The phenomena observed in the study of electricity and magnetism are but the tip of the iceberg.
When Oersted discovered that an electric current in wire affected a compass needle, he missed the point. The magnetic effect is a clue to something far more fundamental to our understanding. The pertinent question is "how is it that the movement of an electric current within a wire can generate an effect outside the wire". The accepted answer that it generates a magnetic field which flows out into the surrounding space is in a sense wrong. What is actually happening is that the electric fields which surround the individual electrons extend out into the surrounding space and move with their electrons. This gives the electric current the ability to generate a magnetic field beyond the wire.
When an electron and a proton form an atom of hydrogen, we find that we cannot measure an electric field of either from a distance. We have a theoretical method of measurement of an electric field by measuring the force on a small test charge. In practice this is impossible, but we can still perform a thought experiment. If we have one proton, it is surrounded by an electric field pointing outward. If we have one electron, it is surrounded by an electric field pointing inwards. When we put the two together, the result is that our test charge no longer shows any signs of experiencing a force. There are two possible reasons for this. Either the test charge experiences two equal and opposite forces, or it experiences no force. Either the two electric fields coexist in space, or they combine to eliminate each other. The accepted view is that they combine to eliminate each other.
The accepted view has its good points. It explains how the energy in the electric fields of the two charges in the atom is less than it would be if they were separate. Thus it provides a storage mechanism for the potential energy which must be given to the two charges in order to free the electron from the proton. But we might well ask if potential energy has to be physically stored. Might it be equally valid to say that potential energy exists by virtue of the geometry of the situation. What I would like to suggest is that the magnetic field generated in the region surrounding a current carrying wire is prima facie evidence for the fact the electric fields of individual charges coexist. They are like invisible fingers reaching out into space from every single charge. When the charge moves, the electric field moves with it. When an electric current flows in a wire, there is an imbalance in the random thermal velocities of the conduction band electrons. This is felt in the region beyond the wire in the relative movement of the coexisting electric fields of the conduction band electrons and of the other charges of the crystal lattice of the wire. It is this relative movement of the electric fields which generates the magnetic field.
The nature of magnetic flux
The relationship between moving charges and their magnetic fields can, I believe, be best understood in the following terms.
Electric fields in relative motion to each other have a property which is called magnetic intensity and is represented by the vector . Every moving charge, that is every charge because they are all moving all the time, has by virtue of the movement of its electric field an associated magnetic intensity field. This is a mathematical artefact of the movement of the electric field and has no real existence. The magnetic intensity at a point is the sum of the individual magnetic intensities of all the charges in the universe at that point and we can express this in the equation.
This sums up the combined effects of relative movements of electric fields and the effect of this combined relative movement is to encourage the formation of a magnetic field. We are accustomed to describe magnetic fields by their magnetic induction . We normally write the equation
but I would modify this to
where the sign is read "would like to be equal to".
The vector has been a convenient descriptor for magnetic fields being defined in terms of the ability of the field to exert a force on a moving conductor. However it has served more than anything else as a device for enabling us to avoid wrestling with trying to understand the nature of magnetic flux.
Magnetic flux is first and foremost a form of energy. It is an energy density flux. It cannot be created or destroyed except that at the surface of a charge, it may be generated or adsorbed changing to and from mechanical energy strictly in accordance with the law of conservation of energy. It is subject to the very severe limitation that it can only move into and out of charges parallel to their electric fields. It has directional properties. These properties are best described by the scalar and vector forms of the magnetic energy density.
We as humans are right at the limits of our ability when we try to comprehend the nature of magnetic flux. It is no surprise that we need to periodically rethink our concept of it. We kneed a stating point and so I am going to use an analogy.
The two properties of magnetic flux; energy density and magnetic induction have the same relationship to one another as do the tilth and the furrows of a field. That is a perfect analogy for anyone with a rural background, but may need a little explaining to others. You start with a plain uncultivated field. Below the grass is soil. You plough the field to create a layer of tilth in which to plant the crop. The plough leaves a geometric pattern of ridges running the length of the field which are called furrows. This tilled soil or tilth as it is called is both created by the ploughing and is the substance from which the pattern of the furrows is made.
When we consider a magnetic field, we need to be aware that it has these two properties of tilth and furrows. The tilth is the energy content of the magnetic field and the energy density might be thought of as the depth of tilth. The furrows are the magnetic induction which we characterise by drawing lines of force. The formation and behaviour of the magnetic field is governed by the properties of both tilth and furrow. At any instance, the lines of force must form continuous loops.
The process by which magnetic energy density flux adds and subtracts is most peculiar and can only be understood in terms of the directional properties of the magnetic field. This is rather technically demanding and is discussed in all its mathematical detail within the section on accelerating a charge.
The observed form of magnetic fields depends entirely on the scale upon which we observe them. From close up, all we see is the individual magnetic fields which surround each charge. The intensity of these fields is many orders of magnitude more than that of any magnetic field we might create in a laboratory or in an electric motor. This field forms circular loops about the line of motion of the charge. At this scale we can use the term "field of motion" to describe the magnetic field surrounding the charge. As we move away from an individual charge, the strength its magnetic field falls off until we find it merging into and combining with the fields of other charges. Eventually we reach human scales and see magnetic fields as revealed by iron filing patterns. But to gain an understanding of the nature of magnetism, we need to consider a simple moving charge. Put aside your concept of an electron and consider a fictional entity which I call a pure charge. It has no mass, no angular momentum and no intrinsic magnetic moment. It is simply a hollow spherical surface of charge.
Imagine we find our charge moving some distance away from any other charge. Nevertheless, the region it is moving through is permeated by the presence of the electric fields of a universe full of charge. It is against this background presence that we measure the velocity of our charge, because it is its relative motion to that background presence which generates its magnetic field. We will consider simple changes in the motion of the charge. First let us accelerate the charge in the direction it is moving. We are increasing the magnetic field and its energy content increases. The magnetic energy density is becoming greater. This means that magnetic energy flux has to be created at the surface of the charge and move outwards. If we calculate the rate of increase of energy in the field of motion of the charge, we can then use the idea that there must be a continuity of presence and movement of the magnetic energy density flux, to calculate the speed with which energy density flux is emerging from the surface of the charge. At this level, we can apply Faraday's law using the velocity with which the energy density flux is moving from the surface. We get the correct answer. That is to say that the force we get is the force required to do the mechanical work to increase the energy content of the field of motion of the charge.
The mathematical analysis can be done for an acceleration in any direction. We find that acceleration is always resisted by a force which is proportional to the acceleration. The other two factors governing the magnitude of the force is the magnitude of the charge and its radius.
The first important result of the proof is that the property we call inertia exists for a pure charge and can be wholly accounted as an electromagnetic interaction.
The second is that starting from a modified understanding of the nature of magnetic flux, we have taken the laws of Biot-Savart and Faraday and proved that Newtons second law of motion, , applies to a pure charge. With this, the laws of mechanics can be deduced.
Faraday's law revisited
It is hard for us imagine how difficult Faraday's original experiment must have been for we are familiar with sensitive moving coil galvanometers. These however did not exist. The galvanometers of the day relied on electrostatic forces and were very insensitive. The famous experiment of Faraday in which a magnet is moved into and out of a coil producing a voltage in the coil has a sequel in which the coil is moved over the magnet and then removed. These two experiments led Faraday to conclude that it was the relative motion between the magnet and the coil which generated the voltage. This concept of relative motion is fundamental to the accepted theory, but it hides from us the real process.
I want to look at the situation where the magnet is moving. In the absence of any coil, we are quite happy to calculate that the moving magnetic field generates an electric field. This is consistent with the theory of the propagation of electromagnetic radiation discovered by Maxwell. The electric field results in an electric polarisation of space. That results in a displacement current flowing which in turn generates a magnetic field which in turn..... And the result is a travelling electromagnetic wave. What is not quite so obvious is the fact that if we place a test charge in the way, it will feel the electric field induced by the motion of the magnetic field pulling in one direction while it finds that the polarisation of space pulls it in the opposite direction with an equal force. So the movement of the magnet should have no effect, but it does.
The answer is far more subtle. The electric field of a charge extends throughout all space falling off in intensity according to the inverse square law. As it moves, the effect of its motion is felt everywhere because of its contribution to the magnetic intensity. We have defined magnetic intensity as and we know that the magnetic energy density is given by . We also know that . When we substitute these equations into each other, we get a most interesting result.
The significance of this result is that the magnetic energy density at a point in a magnetic field is equal to the sum of the energy densities of the individual fields of the individual charges. These contributions can be positive or negative depending on whether the motion of the individual charge is contributing to the magnetic field or lessening it at that point. Now magnetic energy density flux can only move within the confines of the electric fields of charges. If we assume that each charge has its own personal energy density field which is influenced by the prevailing magnetic induction, we find that variations in the magnetic field around a charge will cause magnetic energy density flux to move into and out of the surface of the charge as its position relative to magnetic fields changes.
Let us try to imagine a very well defined situation in which a region of strong magnetic field moves towards the charge. So long as the charge remains outside this region of magnetic field, the energy content of the magnetic energy density flux belonging to the charge remains fairly constant. As soon as the charge moves into the region, there is dramatic change because the energy content now depends on the distance of the charge from the boundary of the region. The charge is moving away from one boundary towards the other and that means that magnetic energy density flux is flowing out of the charge on one side and into it on the other. It is this process which generates the force on the charge. Now the effect on the charge is opposite to the effect of the charge on the field. That means that if the motion of the charge relative to the field is such as to have no effect on the field, then the field has no effect on the charge. The more the motion of the charge affects the field, the greater the force on the charge.
A charge does not have to be within a magnetic field to be affected by it when the field is varying in size and/or strength and/or orientation. The transformer relies very much on this principle. Conventional wisdom has it that magnetic flux moves outwards from the turns of the primary coil into the core as magnetic field builds, then moves back into the turns as the field collapses. In this process, it is said to cut all the turns of wire both of the primary and of the primary coils. This in fact does not happen! There are two factors which contradict this picture. The first can be seen if we imagine a loosely wound coil and draw lines of force for increasing currents. What we find in our two dimensional picture of the expanding magnetic field is that there are zero points between the turns of the coil. The lines of force cannot pass these points. What happens is that lines of force moving towards these points thin and appear to part to join with other lines on the far side of the point. The result of this is that local to each wire, we only find magnetic flux moving outward from the wire. We do not see the flux of the whole field cutting the individual turns as the field changes. The second thing is that within the magnetic core, the process of magnetisation and demagnetization is accomplished by movement of the boundaries of the individual magnetic domains of the material. This involves individual electron orbits changing orientation so that their flux links with neighbours on one side rather than on the other. If we were to attempt to draw diagrams of the movement of the lines of force during this process, we would again find the movements to be local. The mass movement of magnetic flux just does not happen in the way the theory imagines. There has to be a second mechanism which accounts for the induced emf in the coils. It is the process I have described in the previous paragraph.
What the reader needs to understand is that analysis of the workings of a transformer are based on a model which is consistent with energy conservation. It is this consistency which enables the theory to supply answers which are the same as the observed behaviour of the components. The actual mechanism which I have described is also consistent with the conservation of energy and so accounts for the match between theory and practice.
The intimate connection
The process which I have described above becomes more important as we reduce the scale on which we look at a group of charges. Let us imagine that we have a stream of moving charges forming an electric current. The combined effect of the individual movements of the electrons forms the electric current and this generates a magnetic field. We might think of a long thin wire as carrying the current and we find that the surrounding magnetic field has a given energy content per unit length of the wire. If we now stop one electron dead in its tracks, we find that it resists being stopped exerting a force and transferring energy to whatever stopped it. Now the energy which the electron transfers comes from two sources. There is the energy stored in the magnetic field surrounding the electron and we can think of this as the kinetic energy of the electron. Then there is the energy which the electron contributed to the magnetic field which the current generates. But the removal of one electron from the group would mean that the the current is reduced and the surrounding magnetic field needs to shrink. This loss of energy from the magnetic field results in equal amounts of energy have to be transferred to each electron of the current. This flow of energy into the surfaces of the individual charges results in the generation of a force tending to accelerate each electron in the direction of the current. I call this the intimate connection because it links the motion of all of the electrons which constitute the current. A collision which decelerates on electron accelerates all the other electrons. This is the essence of the effect which we call self inductance.
If we come down to the scale of an atom, we find that this process continues. The motion of the electrons is totally chaotic, but basically takes the electrons along sections of curved paths. I call this process pseudo orbiting. The process of moving in an arc should be accompanied by the field of motion of the electron rotating with it, but this process is limited because the further we are from the magnetic field, the faster the magnetic field would have to move. We reach a region where the magnetic energy density flux is literally left behind. If we had a single electron orbiting a nucleus, then the result would be the establishment of a stable orbital magnetic field. Let us imagine that we had one electron orbiting a helium nucleus so that we can imagine a second electron being captured by it. We now have an extremely complex situation in which the resulting orbital magnetic field is composed of energy from both electrons. The electrons interact producing chaotic pseudo orbiting and all the time, magnetic energy density flux is moving into and out of their surfaces generating forces on them. If the change in motion of the electrons is such as to increase their common magnetic field, then the forces on each electron as the magnetic field attempts to grow will tend to lessen the growth in the field, but then if their change of motion is such as to decrease their common magnetic field, the forces generated by the movement of magnetic energy density flux into and out of their surfaces will oppose this tendency. What we can say is that fluctuations in the magnetic fields within and around the atom provides a mechanism for energy exchange between the electrons which has an additional effect on their motion.
What we will find when we attempt to model this and to try to understand the chaotic behaviour of electrons within an atom is only to be guessed at. Will we be able to account for the atomic spectra. Is it possible that some brilliant mathematician will produce phase space models of the chaotic behaviour identical to the present probability distributions of quantum mechanics? One thing is certain: there is more to magnetism than we ever imagined.
© Copyright Bruce Harvey 1997.