Abstract: The Lorentz force law is fundamental for electromagnetism. However, it is known long ago that the Lorentz forces between two current elements do not respect the Newton's third law. This seemingly harmless flaw had never been corrected. In
physical sciences a discrepancy often hides in it new understanding or unexpected breakthrough. For solving this problem, we give a purely theoretical derivation of magnetic force which respects the Newton's third law in the case of current elements and
is identical to the Lorentz force in the case of coils. This new law reveals how electric force is transformed into magnetic force by velocity and is supported by experimental evidences that we will explain and compute with the new law.

1. Introduction
Lorentz force is a fundamental magnetic force which is created by magnetic field on current carrying wire. Let dIa and dIb be current elements and Ba and Bb the magnetic field they create. Lorentz forces on dIa and dIb are dFa and dFb. . Because the
Lorentz forces dFa and dFb are action and reaction forces, they should obey the Newton's third law and sum to zero. However, the Figure 1 shows a case where dFa is perpendicular to dFb , so, dFa + dFb  0, that is, the Lorentz forces that the two
current elements act on each other violate the Newton's third law.

This problem was known for longtime. People justify that the Lorentz forces that two closed loop currents act on each other do satisfy the Newton's third law. Nevertheless, breaking the Newton's third law does not fit scientific standard, even for the
Lorentz forces law which is fundamental. The reason for this problem is that being an experimental law Lorentz force law can only describe forces that experiments show. So far, all experimental magnetic forces are perpendicular to current, so the Lorentz
force law does not describe magnetic force parallel to the current and consequently cannot respect Newton's third law.

We will try to solve this problem with a new magnetic force law that we have derived with pure theory. This new law respects the Newton's third law in the case of current elements and is identical to the Lorentz force law in the case of coils. This law
reveals how electric force is transformed into magnetic force by velocity and is supported by experimental evidences which we will present and compute with the new law at the end.

The new law is derived from the Coulomb’s law which defines the Coulomb’s force for fixed charges. For moving electrons, the Coulomb’s force undergoes relativistic effects and varies with velocity. Although this velocity is very small, the number
of free electrons in wires is so huge that relativistic effects show up nevertheless. We have found two relativistic effects in currents: the relativistic dynamic effect and the changing distance effect.

2. Relativistic dynamic effect
• Couple of charges
Currents flow in neutral wires which contain the same quantity of free electrons and positive charges. For computing the Coulomb’s force in neutral material, we use couple of charges. Let - represent a free electron and + a fixed positive charge
which is a proton in the nucleus of an atom. A couple of charges is represented by
(+, - ) and is neutral. So, the interaction between two couples of charges represents the magnetic force between two neutral materials.

Let (+a, - a) be a couple of charges in the material a and (+b, - b) that in the material b. The vector distance between (+a, - a) and (+b, - b) is r, see Figure 2 (a). Between (+a, - a) and (+b, - b) there are 4 interactions :
+b +a, - b +a , +b  - a , - b  - a . We show them in the line 1 of the Table 1 and in the Figure 2 (b) and (c). We suppose that +a and - a are at the same location and +b and - b are at the same location. Then, the vector distances
between +b and +a, - b and +a , +b and - a , - b and - a are all r .

…

The electric charge of + is e and that of - is -e , then for the 4 interactions the products of charges q1· q2 equal e2, e2, e2 and e2 respectively, see the line 2 of the Table 1. Applying e2, e2, e2 and e2 to the Coulomb’s law (2) we get the 4
Coulomb’s forces which are labeled as F++, F-+, F+- and F- - , expressed in the line 3 of the Table 1 and shown in the Figure 2 (b) and (c).
…

• Dynamic force across moving frames
The sum of the 4 forces equals zero: F++ + F-+ + F+- + F- - = 0. Let us see how the relativistic dynamic effect breaks this equilibrium. In the Figure 3 we have a stationary body b1 and a moving body b2 . The frame 1 is attached to b1 and the frame 2 to
b2. The velocity of b2 relative to b1 is v, so the frame 2 moves at the velocity v in the frame 1.

…

3. Changing distance effect
• Distance change
The second relativistic effect is caused by the position change of electrons. For example, in the Figure 4 the electron moves from - to -’ . Because r is different from r0 the Coulomb’s force it acts on the positive charge + varies from F0 to F. In
the couples of charges (+a, -a) and (+b, -b), the electrons -a and -b are moving and change constantly positions, so the resultant Coulomb’s force that (+b, -b) acts on (+a, -a) changes too. We call this change of the resultant Coulomb’s force the “
Changing distance effect”.
…

4. Complete magnetic force
• Magnetic force on couples of charges
The complete magnetic force that the couple of charge (+b, -b) exerts on (+a, -a) equals the sum of the magnetic force due to the relativistic dynamic effect given in (22) and that due to the changing distance effect given in (43).

…
• Magnetic force on current elements
Let (+a, - a) be one couple of charge in dIa and (+b, - b) one couple of charge in dIb . One (+b, - b) exerts the magnetic force Fba on one (+a, - a). A current element contains a huge number of couples of charges. Let m be the number of couples in
dIa, then one (+b, - b) exerts on dIa the magnetic force m·Fba. Let n be the number of couples in dIb, then n couples exert on dIa the magnetic force n·m·Fba. So, the total magnetic force that dIb exerts on dIa is, see (45) :

…

5. Consequences
• The relationmu0 eps0 c2 = 1
Historically, the values of mu0, eps0 and the speed of light c were measured experimentally. It was James Clerk Maxwell who noticed that mu0eps0c2 = 1 . So, it was an empirical law.

In our derivation this relation emerged naturally from both relativistic dynamic effect and changing distance effect. So, we have theoretically proven this relation and in consequence, the relation mu0eps0c2 = 1 is now a theoretical law.

• Biot–Savart law
The equation (58) is identical to the Biot–Savart law (59) but is derived with pure theory. So, the Biot–Savart law becomes a theoretical law too.

• Lorentz force law
(61) is the Lorentz force that one dIb exerts on dIa . So, we have derived the Lorentz force law from the Coulomb’s law.

• Magnetic force vs. Newton's third law
We have explained in the introduction that the elementary Lorentz force law violates Newton's third law. Let us compute the sum of the Lorentz force that dIb exerts on dIa and the back Lorentz force that dIa exerts on dIb .
The back Lorentz force is obtained from (62) by replacing r0 with -r0 , dIa with dIb and dIb with dIa. Adding (62) with (63) we obtain the sum dFa + dFb which is not zero .

In the contrary, the sum of the magnetic force (49) and its back force is zero. The back magnetic force for (49) is obtained from (49) by replacing r0 with -r0, dIb with dIa and dIa with dIb.Adding FIba given in (49) with FIab yields FIba + FIab = 0
because all the terms cancel out . So, the magnetic force law (49) satisfies the Newton's third law for current elements .

Why the Lorentz force law violates the Newton's third law for current elements? By comparing (62) with (49) we find that (62) lacks the last term Because all experiments are done with closed loop coil over which the integral of this term equals zero, see
(54), the magnetic force corresponding to this term never appear and does not exist in experiments. Being an experimental law, the Lorentz force law does not describe a force that does not exist and thus, lacks this term. So, it cannot satisfy Newton's
third law.

Thanks to the fully theoretical derivation, the magnetic force law (49) contains the missing last term and consequently, satisfies Newton's third law.

6. Experimental evidences
• My experiments
What is the magnetic force corresponding to the last term ? This force is parallel to the current element dIa; such force has never been detected so far. Since this force did not appear in experiments because the test coils were all closed loop, we have
specially designed experiments with non-closed wire and successfully shown magnetic force parallel to the current.

The first experiment is «Continuous rotation of a circular coil experiment» . The video of this experiment is: https://www.youtube.com/watch?v=9162Qw-wNow . In this video we see a round coil that rotates in its plane. Because the coil is round the
driving force must be parallel to the wire, that is, the driving force is parallel to the current. This force cannot be Lorentz force which is perpendicular to the current. A detailed technical explanation is in the paper «Showing tangential magnetic
force by experiment» .

I have also made a « Circular motor driven by tangential magnetic force » . The video of this experiment is: https://www.youtube.com/watch?v=JkGUaJqa6nU&list=UUuJXMstqPh8VY4UYqDgwcvQ . The technical details of this experiment is: « Detail of my
circular motor using tangential force and the equivalence with homopolar motor » .

• Experiment of wire fragmentation
In 1961, Jan Nasilowski in Poland has carried out an experiment which consisted of passing a huge current in a thin wire. The wire exploded into small pieces. The interesting thing is that the wires were not melted but teared apart by mechanical force.
The Figure 9 is a photograph of the exploded wires which shows that all the small pieces have approximately the same length. Jan Nasiowski has published his result in two papers , which are cited by Lars Johansson in his Thesis .

The magnetic force shown in this experiment is parallel to the current and is strong enough to tear the wire apart. Let us explain this experiment with the magnetic force law (49). Take two current elements from the wire, dIa and dIb , the vector
distance from dIb to dIa is r. Let us compute the magnetic force that dIb exerts on dIa using (52). Because dIb is parallel to r, then and the magnetic force is parallel to the current.

We have drawn in Figure 10 the forces FIba and FIab and the current elements dIa and dIb. Let S be a point on the wire. dIa is on the right of the point S and FIba pulls it to the right; dIb is on the left of the point S and FIab pulls it to the left. So,
the point S is under a tension that tears it. If the tension is strong enough, the wire breaks, which was the result of Jan Nasilowski’s experiment.

A wire is broken at a point by the tension created by the segments of the wire on either side of the point . The magnetic force per unit length is the same everywhere in the wire because the current is constant. The segments of the same length create the
same tension. This is why all the pieces of an exploded wire have approximately the same length.

In consequence, the magnetic force law (49) explains well the experiment of Jan Nasiowski.

7. Conclusion

1. We have derived a new magnetic force law from Coulomb’s law and relativity without experimental data.
2. We have proven theoretically the relation mu0 eps0 c2 = 1which was an experimental law. Now the three fundamental constants mu0, eps0 and c are reduced into two primary constants : 0 and c.
3. The Biot–Savart law and the Lorentz force law are derived with pure theory.
4. This new law has a component of magnetic force parallel to the current. Because the Lorentz force law lacks this force, it violates the Newton's third law.
5. We have found two relativistic effects: the relativistic dynamic effect and the changing distance effect. These effects are the deep mechanism that creates magnetic force from electric force.
6. We have presented two of my experiments that show the existence of magnetic force parallel to the current.
7. Our new magnetic force law explains well the experiment of Jan Nasiowski : for the breaking of the wires as well as for the regularity of the lengths of the small pieces of an exploded wire.
8. These experiments give strong evidences for the existence of magnetic force parallel to the current.

So, the new magnetic force law (49) correctly describes magnetic force. Because the new law gives the same prediction as the Lorentz force law for closed loop currents, it works for electromagnetism as the Lorentz force law. However, the component of
magnetic force parallel to the current is new and shown to be rather significant. So, it could be used as the driving force for new devices.

Since the Biot–Savart law, the Lorentz force law and the relation mu0 eps0 c2 = 1are derived with pure theory, the deep mechanism that transforms electric force into magnetic force is revealed to be the two relativistic effects. Consequently,
electromagnetism is much better understood, which will surely unblock the way to unsuspected physical discoveries.


For more detail of this study please read the complete paper here:
« From Coulomb’s force to magnetic force and experiments that show magnetic force parallel to current»
https://www.academia.edu/106863205/From_Coulombs_force_to_magnetic_force_and_experiments_that_show_magnetic_force_parallel_to_current

Kuan Peng

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