2.1    THE CLASSICAL ELECTRIC AND MAGNETIC FIELDS AND FORCES

In Classical Physics the concept of basic charge exists. The charges are said to generate the Electric and the Magnetic Fields and Forces.

The Electric Field of a point charge is a central one with spherical symmetry that represents an attractive force between opposite charges and a repulsive force between equal charges.

The Electric Field generated by a charge Q is expressed by:

E = KQ/r² a_r                     (bold means vectors)

Where r is the distance to the considered point and a_r is the unity vector in the direction from the charge to the point.

The Electric Force of the charge Q over a charge q is then:

F_E = qE

F_E = KQq/r² a_r

It must be noted that the notion of electrical field is a mathematical abstraction that represent the force that would eventually act over a point with unity charge (q = 1) if it would exist in a specific position of the space.

Classically it is said that the Magnetic Force exists where moving charges appears. The movement must be considered relative to the absolute referential R_A as assumed in Section 1.2 part III. Moving charges generates a Magnetic Field and an interaction with other moving charges results in a Magnetic Force acting between them.

It is important to consider here, linear distributions of charge (following a curvilinear path) and we have:

a) The Magnetic Field generated by some linear distribution of charge is then classically expressed by the Biot-Savart law:

dB = (μ₀/4π)I(dl × a_r)/r²  (bold means vectors)

where a_r is the unity vector in the r direction

To find the complete value of B is necessary to integrate over a path.

It is noted that the notion of magnetic field is a mathematical abstraction that represent the force that would eventually act over an element of unity charge (q=1) and unity velocity (v=1) if it would exist in a specific position of the space.

b) The force of a field B over a differential current element ρdlv = Idl where v is the velocity of the charge element in the direction of dl (which is tangential to the element):

dF_B = ρdlv × B = I dl × B          where B = B(r)

To find the complete value of F_B is necessary to integrate over a path.

It is important to observe here a special fact about the current I. A conductor is formed by a structure of neutral atoms that have both, positive charges composed by the “fixed” nucleus and negative charges composed by “moving” electrons in the same quantity. The current I is defined by the net flux of charges, and so, the value of I does not depend on the referential considered to measure it. It can be said as to be invariant under a change of coordinates. But if I is composed by a flux of only one kind of charges, like a beam of electrons, the invariance is lost. The value of I depends, in this case, on the referential choused.

If we consider a Magnetic F ield B acting over a point charge we have the Lorentz force expressed as:

F_B = qv × B

where v should be the absolute velocity and B is the Magnetic Field.

NOTE

The classical Electric and Magnetic fields verify Maxwell Equations.

EXAMPLE

An important example to us is the Magnetic Field generated by a circular ring with radius R. If we consider a referential centered at the center of the ring and the z axis parallel to the ring axis, it can be determined from the equation in a) that the field along the z axis is determined by:

B(z) = ½ μ₀IR²/(R² + z²)^3/2

If we consider large distances z >>R  then:

B(z) ≈ ½ μ₀IR²/z³   which is proportional to 1/ z³

Here we can see that, at small distances, the Magnetic Field can be stronger than the Electric Field which is proportional to the inverse of the square of the distance.