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.
The Figure 1 shows a case where dFa is perpendicular to dFb , so, dFa + dFb ¹ 0. 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.
We will try to solve this problem with a new magnetic force law that we have derived with pure theory. 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.
· 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
The sum of the magnetic force (49) and its back force is zero. So, the magnetic force law (49) satisfies the Newton’s third law for current elements . 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.
· My experiments
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.
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, electromagnetism is much better understood.
For more detail of this study please read the complete paper here:
 Kuan Peng, 2018, «Showing tangential magnetic force by experiment», https://www.academia.edu/36652163/Showing_tangential_magnetic_force_by_experiment
 Kuan Peng, 2014, Video https://www.youtube.com/watch?v=JkGUaJqa6nU&list=UUuJXMstqPh8VY4UYqDgwcvQ
 Kuan Peng, 2014, « Detail of my circular motor using tangential force and the equivalence with homopolar motor » , https://www.academia.edu/7879755/Detail_of_my_circular_motor_using_tangential_force_and_the_equivalence_with_homopolar_motor