M S La Moreaux

Does anyone really need a diagram of a straight current-carrying wire? Length contraction applies not only to moving objects but also to the spaces between those moving with the same velocity.

Copper contains 10^22 free electrons per cubic centimeter. A paradox within special relativity may be impossible, but that is not the situation I am describing. It is instead a discrepancy between what special relativity predicts and what is actually observed. A magnetic field affects a moving charge, not a stationary one. The effect the I am describing would affect a stationary charge external to the wire.

There is no cutoff velocity of relativistic effect. Although the drift velocity of the electrons is minuscule, the relativistic increase in their density multiplied by the huge number of free electrons is palpable, on a par with magnetism. I do not see that the ladder paradox applies. In the ladder paradox, the paradox involves the discrepancy between the views of two different reference frames. The relativistic length contraction of the ladder, and thus the increased density, as it were, of the rungs in the barn's frame is not denied.

This really should not be this difficult. I assume a straight side to avoid acceleration so special relativity will apply. The circuit is stationary. The only things moving are the free electrons. Their charge density appears greater while the charge density of the protons remains unchanged, thus resulting in a net negative charge for the segment of the circuit under discussion. This should be observed by someone in the frame of reference of the circuit, but is not.

Consider a current carrying circuit with a straight side. Because of length contraction, in the straight side the moving electrons will appear closer together than they would be if there were no current. This gives a greater charge density, so the straight side should have a negative charge. But this is not observed. How come?

I have personally discovered that Faraday’s Law of Electromagnetic Induction is not a law. I am referring to the version of Faraday’s Law which uses the ordinary time derivative and not the partial derivative, which version is called the Maxwell-Faraday Law and is one of Maxwell’s equations. Faraday’s Law is not a law because it is not an expression of a physical principle. There are two physical principles which are called induction. The first is motional emf, which is an extension of the definition of the magnetic field and consists of a conductor moving through a magnetic field such that it cuts the magnetic field lines. It is responsible for the operation of generators. The second is the Maxwell-Faraday Law, one of Maxwell’s Equations, and is responsible for the operation of inductors and alternators. These two principles are independent of each other. It is unfortunate that both are called induction because it leads to confusion. Faraday’s Law only partially includes both and in a way which obscures the cause and effect. Neither of the principles of induction requires a circuit, but Faraday’s Law does. Faraday’s Law gives the same result whether the change in magnetic flux linking the circuit is strictly due to a time-varying magnetic field or is due to movement of the circuit. A time-varying magnetic field is accompanied by an electric field. It is this electric field which results in an emf in the circuit. A moving circuit in a magnetic field experiences an emf due to the motion of the conductor through the magnetic field. Of course, the magnetic field has to be non-uniform or the shape of the circuit has to change with time. The same mathematics is that is used to calculate the time derivative of the magnetic flux linking the circuit is used to calculate the motional emf. Thus, Faraday’s Law is just based on mathematics and not physics. It is misleading because it implies a cause and effect which does not exist. Also, Faraday’s Law can be derived. This proves that it is not a law because physical laws cannot be derived, only inferred. I believe that Faraday’s Law does not belong in the textbooks. It is superfluous and serves no purpose except to confuse the issue.

I have concluded that Faraday's Law is not a law but rather a mathematical identity. If the change in magnetic flux linkage of the circuit is due to motion of the circuit, the resulting emf is just motional emf and is not a result of flux change. Essentially, the quantities that are multiplied together to get the emf are the flux density, the length of the conductor, and the speed of the conductor. These are the same quantities multiplied together to calculate the rate of change of the magnetic flux linkage. Faraday's Law may give the correct result, but it is not the expression of a physical principle, which I believe is a requirement of a physical law. Now, if the circuit is stationary and the magnetic flux changes, an emf will be induced in the circuit, and Faraday's Law will give this as a result. But this case is covered by the Maxwell-Faraday Law, which is one of Maxwell's equations. Between them, motional emf and the Maxwell-Faraday Law cover every case of induction. Faraday's Law is not just a combination of these two principles: Faraday's Law applies only to a circuit. Neither motional emf nor the Maxwell-Faraday Law do. Motional emf only requires a conductor, and the Maxwell-Faraday Law does not even require that.

Someone elsewhere pointed out to me what appears to be the correct answer to my original question. There will be no voltage across the magnet because both types of induction are present and oppose each other exactly. The magnet as conductor is moving in a magnetic field (its own) and producing a motional emf. However, the magnetic field is increasing in the space in front of the moving magnet and decreasing behind it. These changes in the magnetic flux at stationary points of space are accompanied by an electric field which counteracts the motional emf.

I am not sure that the cutting of flux lines occurs in alternators, where the magnet is the moving part. Different frames seem to give different results. For example, imagine a magnet and a length of wire in the magnet's field. If the wire moves and the magnet does not, there will be a non-electrostatic emf along the wire. If the magnet moves and the wire does not, there evidently will be no emf along the wire because magnetic field lines do not move. I would be interested if a source could be found which treats this topic. I have seen a quote by Einstein which points out that the two cases are treated differently even though they would seem to be symmetrical. In the case of the rotating magnet, the idea of the magnet feeling a force from the current-carrying wire seems reasonable. A magnet is supposed to contain spinning electrons. I am having trouble visualizing exactly how they would be affected by the wire's magnetic field, though. The idea that there are no moving magnetic flux lines and the idea that a magnet's field is not attached to it are related to the Faraday paradox. The setup is an electrically conductive disk and a disk magnet mounted face to face on separate axles. There are brushes at the axle and the rim of the conductive disk. If the conductive disk is spun while the magnetic disk is stationary, there is a voltage between the brushes. If the conductive disk is stationary while the magnetic disk is spun, there is no voltage between the brushes. If both disks are spun together. There is a voltage between the brushes. If the voltage were the result of the cutting of a circuit wire by the field lines of the magnetic disk, then there should be a voltage when the magnetic disk alone is spun.

They are not exactly free to move. The drift speed of the conduction electrons is something on the order of a micrometer per second. That is insignificant compared to the typical rotation speed of the disk.

There is a version of the homopolar motor consisting of an electrically conducting disk magnet (whose faces are the poles), mounted on an axle like a wheel, with brushes on the axle and the rim. When an electric current is passed through the disk by means of the brushes it rotates from a standing start. This would seem to indicate that the magnetic field is not attached to the disk.

That is the thing. It seems that the magnetic field is not attached to the magnet. There is evidently no such thing as translating magnetic field lines. There is obviously a field around the magnet no matter where it moves. It is evidently an illusion that the field moves with the magnet. The magnetic flux density could change at any given point over time in such a way that it appears that the field is moving when, in actuality, it is not.

If an electrically conductive permanent magnet is moved at right angles to the direction of its internal magnetic field lines, is a non-electrostatic emf induced across the magnet?

I have realized that my purported counter example involving the toroidal transformer is flawed. The circuit splits at the slip ring and then rejoins. This is not addressed by Faraday's Law. Mike