Bruce Harvey's Alternative Physics site

Errors in the theory of Electricity and Magnetism


aether, Michelson-Morley, flux, magnetic flux, Einstein (1) (2), long thin wire, acceleration, Faraday's law, inductance (1) (2), energy density flux


It seems strange to suggest that anything could be wrong with the theories of Electricity and Magnetism. All the laws are enshrined in equations which are in daily use in the design, construction and testing of electrical devices. There is nothing wrong with the equations, but a little thought about the history of the development of the theory will cast some light. Magnetism is a mystery: at one time it was thought to consist of currents of aether, the all pervading substance of space. This led to the development of vector field theory in which the Grad and Curl operators were perceived in terms of fluid mechanics. Aether theory fell out of fashion after the null result of the Michelson-Morley experiment which was designed to detect the movement of the earth through the aether; but the vector algebra still described electromagnetic phenomena adequately and the disappearance of the aether left our concept of magnetic flux much unchanged. We simply forgot that flux meant flow and came to think of it as something which had bulk. Our understanding of magnetism was not helped when Einstein decided that electric and magnetic fields were not separate entities, but different aspects of the same entity called an electromagnetic field. According to Einstein, the velocity of the observer determines the amount of electric field and magnetic field which he sees.

I do not see how Einstein's concept can explain the storing of magnetic energy in an inductance. It may work perfectly for single fields, but given the thermal motion of the electrons in the conduction band of a wire, we have to have a theory which enables us to explain how the motion combines into the formation of a magnetic field which is sensitive to the geometry of the circuit.

It seems to me that the phenomena of the storage of energy in an inductance can only be explained if we make three modifications to our concepts of electric and magnetic flux.

Magnetic flux must be in the form of continuous loops, but it is not confined to remain in the same loops so that magnetic fields in relative motion to each other are able to merge in a fluid manor. As a compass needle rotates, its flux alternately forms closed loops and then links with the earth's magnetic field, individual stands breaking and immediately re-linking with others to maintain the continuity.

The energy content of a magnetic field can only change by the movement of magnetic energy density flux into or out of the surfaces of electric charges. Electric fields act as guide rails for the movement of magnetic flux where the form of a magnetic field is changing.

Magnetic flux

One of the mistakes made in the Nineteenth Century lay in the interpretation of the nature of magnetic flux. We define the strength of a magnetic field in terms of the force it exerts on a wire carrying an electric current, but because magnetism had been investigated before the the discovery of the interactions between Electricity and magnetism, two measures of the strength of a magnetic field were already in existence. The magnetic intensity H was understood as a force field produced by a magnet. Different kinds of material reacted in different ways to the presence of the magnetic intensity H. The answer was to define a second quantity, the magnetic induction B and link the two by the equation.
where µ is a property of the material the magnetic field is in.

It took some time for the definitive equations to emerge from a number of possibilities into the now familiar

where µ0 is a property of space called the "permeability of free space"and µ is a number called the "relative permeability" of a material.

Faraday discovered that it was the magnetic induction B which was responsible for inducing electric currents in wires which were in relative motion to the magnetic field. He listed three ways in which this relative motion could be generated. The magnet could be moved; the wire could be moved or the strength of the magnetic field could change. Faraday attributed bulk to the magnetic induction field and used the word flux to describe it. The magnetic induction B was a measure of the density of this flux and the total amount of magnetic flux could be found by multiplying the flux density by the area it passed through. Since magnetic fields are seldom uniform, we usually do this by integration.

The long thin wire

Text books usually include a number of examples in which the magnetic field can be calculated. One of these is the long thin wire. A uniform current i is said to flow down a long straight wire which is considered to be infinite in length. The wire is surrounded by a circular magnetic field. A formula can then be deduced by integration for the magnetic field strength at a distance r from the wire.
So far, so good.

The problem comes when we apply Faraday's law of induction to the situation. We start with a wire with no current flowing and try to establish an electric current in it. The magnetic flux which will surround the wire is generated within the wire and moves outward. As it does so, it induces a voltage in the wire opposing our attempt to generate the current. Common sense says that an infinite wire will generate an infinite voltage and we will not be able to establish the current. However, since infinite wires are impossible, it is but a small step to assume that we have infinite voltages available and resolve the problem by considering what happens to 1 metre length of the wire. We should be able to calculate the flux surrounding this and find the induced voltage. But the answer we get is still infinity.

It turns out that this is an unsolvable problem, but in attempting to solve it, we make a discovery of great importance. One of the lines of enquiry is to consider the current as moving electrons. The simplest current of all would be a single electron. The velocity of the electron would generate a magnet field around its line of motion and changes in the velocity of the electron will require magnetic flux to move through the surface of the electron. The resulting electric field will resist the change in velocity of the electron.

Charges that should not move

A charge moving with a velocity v is surrounded by a circular magnetic field
This defines the magnetic induction at any point from the centre of the charge. As the velocity is increased, the magnitude of at each point increases. Faraday's law predicts that when the strength of a magnetic field changes, an electromotive force is generated. To calculate this for a loop of wire carrying a current, we take a plane which is intersected by the magnetic field and carry out
over this surface. This gives us the magnetic flux in the field. At each point on the boundary of this surface, flux is moving across the boundary and this results in an electric field . Integrating around the boundary, we get the Emf predicted by Faraday's law.

For our moving charge, the surface must be any half plane from its line of motion to infinity. We then use polar coordinates .

This infinite result implies that any attempt to accelerate the charge will result in an infinite increase in the magnetic flux which surrounds it. This infinite increase will according to Faraday's law produce an infinite resistance to our attempts to accelerate the charge. The consequence of this is that if the charge starts from rest, it will never move.

Understanding infinities

In the case of the long thin wire, we get an infinite answer because we have imposed an impossible question. We have set up an imaginary situation which has several layers of impossibility and more than one of these is able to produce an infinity. When we tried to get around the question of the infinite voltage required by considering a metre length of the wire, we still have two more impossibilities. The first is the impossibility of applying a voltage to the whole length of the wire at the same time because we can only communicate information at the speed of light. The second is that the change in magnitude of the field can also only be implemented at the speed of light.

By considering only the movement of a single electron, we eliminate the first and hopefully make the second controllable. However, we are left with a problem. If we consider a thin spherical shell surrounding the electron, the strength of the magnetic field within this shell is inversely proportional to the radius squared. The area of the plane intersected by the spherical shell is proportional to the radius so that the net result is inversely proportional to the radius. When we try to integrate 1/r, we get infinity because the function 1/r does not diminishes in size fast enough. We need a function which will diminish in size at a fast enough rate to make its integral finite.

Energy density flux

If we consider the energy stored in magnetic flux, we get an energy density and when we substitute the expression for the magnetic field surrounding a moving electron into this we get:
But energy density exists throughout all space and we must now integrate over the volume of the spherical shell which is proportional to the radius squared. The quantity of magnetic energy contained in the shell is proportional to 1/r2 and we get a function which is integrateable.

If we try to calculate the total magnetic induction flux surrounding the moving electron, we get an infinite answer, but if we try to calculate the total energy, we get a finite answer. The only conclusion to draw from this is that magnetic flux is primarily an energy density flux. Faraday's concept of magnetic flux is fundamentally flawed. As we generate a magnetic field, the thing which has bulk and which moves through space is not the magnetic induction B, but the energy density flux Qm . There are many geometrical situations where there is a good one to one correspondence between the two and in these circumstances, Faraday's law gives good results, but in the case of the electron, it fails miserably. What we have to do is to work in terms of the energy content of the magnetic field. We know the energy density at the surface of the electron in terms of its radius and this allows us to calculate a velocity at which energy density flux is moving into or out of the surface of the electron. It is this which generates an electric field opposing changes in the velocity of the electron.

These results are used in the mathematical treatment of the Pure Charge Model to show how the flow of magnetic energy density flux out of and into the surface of a charge generates inertial forces in accordance with the laws of conservation of energy. Those results are then used to derive Newton's laws of motion for gross matter.

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© Copyright Bruce Harvey 1997.