(1971 — 2 XII 1996, 12 II 2000)

Professor Boryara (Papua New Guinea)

1. Bio-Savard’s Law

The experimental law according to which the segment of the conductor ΔL, through which the current flows with a force of I, creates at a given point in M space a magnetic field with a strength ΔΗ equal to:

ΔΗ = k I Δ Lsin φ / r 2, where

r is the distance from ΔL to M,

φ is the angle between ΔL and r,

k is a coefficient depending on the selected system of units,

I is the current in the conductor

Simplifying the formula for the case when r is perpendicular to L, a segment of a conductor with a length L at a distance r creates a magnetic field with a strength H if the current through the conductor has a magnitude I:

H = k I L ∕ r 2

1. Provocation.

Let’s consider two thought experiments.

A) There is a long vacuum tube with a heated cathode and anode. (Fig.1) The vacuum in the tube is high enough so that the free path of the electrons is much longer than the length of the tube. By applying a potential difference (anode +, cathode -) to the cathode and anode, we create an electric current in the tube of magnitude I.

Let’s place magnetometers along the tube (near it), magnetically shielding each of them from the other, so that each magnetometer M 1,2, 3, i measures the energy created only by the portion of the tube assigned to it ΔL 1,2,3, i.. All segments of ΔL are equal in length. Then, according to the Bio-Savard law, all magnetometers will show the same values of H, because the current I is the same, and the length of each tube segment is the same. I.e. N 1 = N 2 = N 3 = N i ….

But in the first segment ΔL 1, the velocity of the electrons just emitted by the cathode is small, while in the last (at the anode) ΔL of the anode it is many times higher, because the electrons are accelerated by the electric field in the tube. And although the current in the tube is the same, the electron velocities are very different at the cathode and anode. The magnetic field strengths along the tube are the same.

Conclusion: the magnetic field strength created by moving charges DOES NOT DEPEND ON THEIR VELOCITY!

B) There is a dielectric tape with electric charges uniformly fixed on it. The charge density is the same everywhere – p. The length of the tape is L. The tape can move with the help of movable rollers at its ends. (Fig.2)

Near the tape, in the middle, there is a magnetometer that measures the strength of the magnetic field created by charges moving with the tape. All parts of the tape outside its active length L are magnetically isolated and cannot affect the readings of the magnetometer. When the tape moves at a speed of v, electric charges generate a current I = pv, which in turn creates a magnetic field with a strength of N.

Let’s increase the tape pulling speed by n times. Then the current will also increase n times and, all other things being equal, the field strength will also increase n times.

But the problem is that not only the current I has increased n times. But the speed of the charge carriers also increased by the same number of times. So, if the magnetic field strength also depended on the velocity of the charges, it would have to grow not by n, but by n 2 times! But it has grown only n times. Therefore, the strength of the magnetic field does not depend on the velocity of the charges!

However, such a conclusion from these two thought experiments contradicts the fundamental statement that it is the movement of charges that creates a magnetic field and the faster the charge moves, the greater the field generated by it. This can be seen, in particular, from the formula for the Lorentz force acting on a charge moving in a magnetic field.

So, a paradox has arisen.

1. My explanations.

Of course, the conclusions of part 2 are incorrect and there is no paradox! And the “Bio-Savarists” do not contradict the “Lorentzists”.

The root of the “paradox” is in the physically incorrect formulation of the Bio-Savard law.

This law is experimental and has been tested in thousands of experiments and billions of applications in all electric machines. Mathematically, it is also formulated flawlessly and calculations based on it have also been confirmed in experiments and in practice. But its physical formulation is not quite correct and strict physical analysis shows this, leading to the above-mentioned “paradoxes”.

Let’s pay attention to the current – I. Electric current, by definition, is the directed and orderly movement of electric charges. The magnitude (or strength) of this current is determined by the number of charges that have passed through the cross section (conductor) per unit of time – Q ∕ t. Note that the velocity of charges does not explicitly appear in these definitions, although we are talking about movement, displacement of charges. The length of the conductor – L is included in the formula of the law as an unrelated quantity. This creates a “paradox”. Instead of the magnitude (strength) of the current and the length of the conductor, taken separately, in the Biot-Savard law, which is physically correct, there should be “a full charge in the conductor — Q, moving at a speed v relative to the point (M) at which we observe a certain magnetic field with a strength of N.” In this formulation, the Bio–Savard law is not very convenient for calculations, it sounds somewhat cumbersome, but it is physically correct and removes these “paradoxes”.

So, the law should be formulated as follows:

The strength of the magnetic field at point M depends on the magnitude of the total charge Q moving at a speed v relative to this point at a distance r perpendicular to v. (If not perpendicular, we introduce the sine φ into the formula.)

Let us now apply this “new-old” definition of the Bio-Savard law to the above-described experiments. Let’s start with the second one.

There is a total electric charge Q on the entire plane of the tape, equal to the product of the surface charge density p by the area S, i.e. pS (in a volumetric conductor, p is volumetric by V is volume). The density of charges on the tape does not change from its movement (the tape is rigid and does not stretch), and the length of the tape remains constant too. By changing the speed of the tape pulling, we only change the speed of movement of the full charge Q relative to the point M and, accordingly, the magnetic field strength at this point changes proportionally. No “square n” appears here (we have replaced “current strength and conductor length” with the terms “full charge and its speed”)! The “paradox” has been removed.

In the first experiment, for each selected tube segment ΔL 1,2,3, i, both parameters change: both the full charge Q i and its velocity v i. Since in the flow of electrons accelerated in the tube, the charge density changes linearly depending on the velocity (with an increase in velocity by n times, it also decreases by n times), the product Q i v i remains constant for all segments of the tube, which means that the magnetic field strength induced by them also remains const. The “paradox” has been removed.

Mathematical perfection is not physical perfection! The formula of the Biot-Savard law includes the product I x L . The following mathematical operation is often performed: I = Q/t at the same time, substituting the result into the primary formula, IL = QL /t is obtained and, since L /t is the speed (distance divided by time), IL = Q v is obtained, i.e. the product of the charge by the speed at which He’s moving. The physical meaninglessness of such a transformation (in which the right part has a clear physical meaning, and the left part is meaningless) becomes clear from the following reasoning:

Firstly, L is not a distance, but the length of the wire and L/t is physical nonsense – “the length of the wire per unit of time” – although in dimension it is speed, everything is mathematically correct!!!

Secondly, let’s assume that the current is 1 ampere (i.e. one coulomb per second), and the length of the wire is 1 meter. One pendant is 1.6 x 1019 electrons and, following the above transformation, we obtain the velocity of movement of these electrons (clouds of electrons) – 1m/s. If we increase the length of the wire 10 times, the electron velocity will also increase 10 times and become 10m / s, while the true electron drift velocity depends on the electric field strength inside the wire and if the current is still 1 a, then the voltage remains the same! In addition, the physical drift of electrons in a conductor (copper wire) is incomparably slower than meters per second! In copper wires, it is generally impossible to obtain a field strength greater than 0.1v/cm. ” Such tension already gives a maximum technical current load of 6 a/mm2. Nevertheless, a swarm of electrons at the same time has a moving velocity of only 0.4 mm/sec in the wires!!!” (Quote from R.W. Pohl, The Doctrine of Electricity, p. 427)

And besides, if you increase the length of the wire to 300,000 km, it turns out that electrons drift along the wire at the speed of light!!!

In reality, it will be like this: I = Q/t = p el V/t, where p el is the volume density of electrons in the conductor, and V is the moving volume of electrons in the conductor equal to v electrons x S, where S is the cross—sectional area of the conductor, and v el is the velocity of electron drift in the conductor. Then I = p el v el S/t; and IL = p el v el SL/t, where SL is the volume of the entire conductor with length L and cross-sectional area S. Replacing SL with V pr (i.e. the full volume of the conductor), we write down the Bio-Savard law in the form

H = kIL/r 2 = k p el V pr v el/r 2.

In such a formula, the relationship between the magnitude of the current and the rate of electron drift along the wire is obvious and the paradox is removed. It also includes the total charge in the conductor — p el V pr and the drift velocity of this charge — v el in it.

 Summary: The Bio-Savard law in its classical form is not physically correct and must be expressed in the form given on pages 3 and 4.

 From the collection of articles by Professor Boryara: “The Involution of Physics”

12 X 2004

Addition to the article.

The problem, as I have already pointed out, is actually not in the Bio-Savard law itself.. It lies in an incomplete or physically incorrectdetermination of the magnitude of the electric current!

If the current value is defined not as “an electric charge Q flowing through a cross-section per unit of time”, but as“the number of electric charge carriers with a total chargenqdistributed uniformly in volumeVand passing at a speedUthrough the cross-sectionS.“, then

in this form, the law is not contradictory.

It turns out to be cumbersome, but physically true.