# 9 Webmaster & Author: Antonino Cucinotta  ## THE ELETTROMAGNETIC (LORENTZ'S) FORCE BETWEEN AN ELECTRIC CHARGE AND A MAGNETIC FIELD An electric charge, moving with the speed v in a magnetic field having the intensity H, is subjected to the Lorentz force, which acts along the perpendicular to the plane individuated by the lines of force of the magnetic field and by the motion line of the charge (the instant
direction of its speed ), with an intensity given by the formula:
F = kqvH, where k is a constant of proportionality depending on the medium which the charge is moving inside, and on the measure unit system which is used (for example, in the international system M.K.S.A. and in vacuum, practically also in the air,k coincides with the so-called vacuum magnetic permeability mo = 4 p 10-7 Henry/meter ).
The force of Lorentz acts on both the electrons moving inside the electric circuits (the electrons constituting the electric current) and on any charged particle moving in the air or in vacuum.
If a charged particle is moving with a velocity perpendicular to the lines of force of the magnetic field, the force of Lorentz constitutes the centripetal force that is continously diverting the particle from its instant motion line, in such a mode that its trajectory is a circumference placed in a plane perpendicular to the lines of force of the magnetic field.
Instead, if a charged particle is moving along a direction which is oblique with respect to the lines of force of the magnetic field, the component of its velocity along the perpendicular to the magnetic field direction, determines the intensity of the Lorentz force, while the component of its velocity along the magnetic field direction,makes the plane of the circular trajectory be shifting with a constant speed parallel to itself .
Therefore, as a consequence of the Galilean principle of the motion composition, the charged particle describes an helical (spiral) motion having its axis along the magnetic field direction.
Some examples:
The flux of the particles constituting the solar wind (high energy protons and electrons) interacts with the Earth magnetic field that, by means of Lorentz's force, makes them describe spiral trajectories along the lines of force directed from the magnetic North pole toward the South one. Therefore electrons and protons are trapped by the Earth magnetic field giving rise to two radioactive zones (the Van Allen zones), that were discovered in 1958 by means of the first USA artificial satellite of the "Explorer" series.
The Van Allen zones give rise to a natural biological screen that attenuates strongly the flux of the solar ionizing particles investing the terrestrial surface. The interaction of the particles of the solar wind with the terrestrial magnetic field is accountable for the polar auroras that are observed in coincidence with notable periodical increases of the solar activity (sunspots and magnetic storms ).
In the circular accelerators for elementary particles ( cyclotrons, electro-synchrotrons, proto-synchrotrons and storage rings ), the force of Lorentz produced by the interaction of the charged particles with intense magnetic fields, determines the radius R of the circular trajectory and then the dimensions of the accelerator,which depends, with an assigned speed v, on the mass m, on the electric charge q of  particles and on the intensity H of the magnetic field:
(centripetal force = mv2/R ) = ( the force of Lorentz = kqvH);
R = mv/(kqH), where k is the proportionality constant of the Lorentz law .
Therefore are necessary very strong magnetic fields, produced by superconducting magnets maintained at very low temperatures (a few Kelvin degrees) to reduce the dimensions and the cost of the accelerators, compatibly with the very high energy ( 1000 Gev ) that must be reached.
The circular accelerators operating at the CERN of Geneva and at the FERMILAB of Chicago have circumferences of several kilometers .

## THE ELECTROMAGNETIC FORCES BETWEEN AN ELECTRIC CIRCUIT AND A MAGNETIC FIELD

The force of Lorentz that acts on the electrons constituting the electric current in a conducting wire, determines the macroscopic force acting on the conductor when it is placed in a magnetic field.
The magnetic force F = k H L I is directed perpendicularly to the plane individuated by the wire and by the direction of the magnetic field, and its intensity is directly proportional to the current intensity I, to the length L of the wire and to the intensity H of the magnetic field ( k is the proportionality constant of the Lorentz formula).  The transformation of electric power into mechanical energy in an electric motor is possible as a consequence of the fact that the electromagnetic forces acting on the conducting wires of the rotor winding by means of the magnetic field produced by the stator, produce a system of torques, that are adding each other to give the engine torque generated by the electric motor.

## THE ELECTROMAGNETIC FORCES (ELECTRODYNAMIC FORCES) ACTING AMONG ELECTRIC CIRCUITS

Even the macroscopic electrodynamic forces acting among the conductors of an electric circuit (for example among the coils of the winding of an electromagnet, an electric motor, a dynamo, an alternator or a transformer) are owing to the microscopic Lorentz forces to which  are subjected the conduction electrons.
That forces can be always reduced to a force and a torque, which are described by Ampere's electrodynamics laws .
In the particularly simple case of two rectilinear and parallel conducting wires of length L, in which are flowing the currents with the intensities I1 and I2, placed in vacuum (practically also in the air ) at the distance d, the electrodynamic force acting between them is directly proportional to the product of the current intensities I1 and I2 and to the length L, and is inversely proportional to the distance d:
F = k I1I2L/(2pd), where k is the proportionality constant of  Lorentz's law. If, in particular, in both the conducting wires flows the same current (coils of a winding ), electrodynamics forces depend on the square of the current intensity.

## MAXWELL'S  ELECTROMAGNETISM

The Ampere law of the magnetic concatenation was generalized by Maxwell, who, exclusively on the base of physical-mathematical elaborations, deduced that a varying magnetic field may be produced, besides by a current flowing in an electric circuit, even by a varying electric field in vacuum, even if there are neither electric charges nor electric circuits.
For analogy with the magnetic field generated by electric charges moving in vacuum and by electric circuits, Maxwell introduced the so-called "displacement currents" in dielectric (insulating) materials, that are produced by varying electric fields.
Therefore, as exist magnetic lines of force concatenated to an electric circuit, so exist magnetic lines of force concatenated to a varying electric field.
Maxwell, on the basis of the fundamental principles that govern the electric and magnetic phenomena (the Coulomb's, Ampere's and Faraday-Neumann's laws), succeeded to elaborate his theory of electromagnetic phenomena, furnishing a genial and elegant unitary representation of both electric and magnetic phenomena, by means of four fundamental equations (Maxwell's equations ), that can be expressed both in a differential form relating to infinitesimal spatial zones in which take place electromagnetic phenomena, and also in an integral form, more comprehensible and suitable to describe the electromagnetic phenomena that are observed in laboratory.
The first equation, equivalent to the Gauss theorem, can be obtained on the basis of   Coulomb's law, and it expresses that the flux of the electric field (number of the lines of force of the electric field ) coming out of  a closed surface containing one or several electric charges, is directly proportional to the algebraic sum of all the positive electric and negative charges, which are placed at rest in assigned points of the space inside the considered surface.
For example, if we consider a spherical or a cylindrical surface containing some electric charges , it is possible, by the first Maxwell's equatio, to calculate the intensity of the electrostatic field in the points of the space inside and outside the considered surface.
The second Maxwell's equation is relating to the characteristic property of the magnetic field, whose lines of force are always closed, since don't exist any single magnetic poles .
Therefore, if we consider a closed surface in which are placed some magnets or electric circuits, the number of the magnetic lines of force  coming out of the surface is always equal to the number of the ones going into the surface, since the lines of force that have origin inside the surface go back always into it ending in the same points from which they originated.
The third Maxwell's equation corresponds to the electromagnetic induction law (Fraday-Neumann's law), that implicates always the existence of a varying electric field along a closed line (an electric circuit or a closed line traced in vacuum) crossed by the lines of force of a time changing magnetic field.
The fourth Maxwell's equation corresponds to the generalization of  Ampere's law to the the magnetic fields generated both by electric charges moving in vacuum, inside liquids, gases and conductors (that is by electric currents ), and also by "displacement currents", that is by time changing electric fields.
Therefore there is always a magnetic field concatenated to a closed line, inside matter or in vacuum, which is crossed by moving electric charges (electric currents) or by the lines of force of a time changing electric field.
The most important result of Maxwell's electromagnetic theory consists in the fact that the fourth equation of the electromagnetic field implicates the existence of electromagnetic waves that are propagating in vacuum with a speed that equalizes the speed of light
(300000 Km/s ), transporting energy and momentum of electromagnetic kind.
As a consequence of this result Maxwell enunciated the electromagnetic nature of all the luminous phenomena (the so-called electromagnetic theory of light).
Her intuition was brightly confirmed in the ending period of XIX century by means of the bright experiments effected by Hertz on the electromagnetic waves ( Hertzian waves ).
Hertz succeeded to produce by electromagnetic waves reflection, refraction, interference, diffraction and polarization phenomena analogous to the ones produced by the luminous waves.

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