This short article wants to be a homage to one
of the greatest geniuses of electromagnetism, whom we, electrical technicians
and engineers, owe something: James Clerk Maxwell.
The
classical theory of electromagnetism, that we study today in the formalism
developed by Hertz, is summarized in the Maxwell’s equations. Or, more properly, we can say, paraphrasing
Hertz, that the Maxwell’s equations are the electromagnetism.
A treatment
of electromagnetic phenomena could be accomplished without Maxwell’s equations,
but unfortunately, such a treatment would encounter huge difficulties, because
the expressions of the electric field e and of the magnetic field (or,
better and more properly, of the magnetic induction b) are very complex, in
the case of particles in a generic motion. For this reason, in the macroscopic
world, it is much more convenient to treat the problems of electromagnetism through
equations which establish relationships between the vectorial fields e
and b.
These relationships are in fact the Maxwell’s equations. They represent a
synthesis of the properties of electromagnetic fields and allow, together with
the boundary conditions, to handle all problems of electromagnetism in the
macroscopic world, with applications ranging from electrical motors and
generators, to circuit breakers, to telecommunications and many others.
The
original idea of Maxwell was to apply to electromagnetism the equations and the
analytical methods used by Lagrange to establish the relationship between potential
and kinetic energy, being convinced that electrical phenomena cannot be disconnected
from magnetic phenomena. In his memory “A
dynamical theory of electromagnetic field”, presented to the Royal Society
of London in the year 1864, he deducted the system of equations in which
electromagnetic quantities show analogies with their mechanical corresponding
entities.
Let us try
to understand the meaning of these equations.
First, they
give the model previously developed by Faraday (based on the concept of lines of force) a mathematical form. The
concepts of field and of line of force are introduced in books of physics and
electrotechnics as pure mathematical entities, which acquire physical content
only when in a point of the field there is an electrical charge. In other
words, the idea that the action between two electrical charges is direct and
immediate, without any impact of the surrounding space, appears fully justified
and plausible. This is the way how the Coulomb’s law presents the problem,
concentrating the attention on the charges.
An
independent treatment can be developed drawing the attention on the properties
of the field and deducting the actions among electrical charges as consequence
of these properties, so simply reducing the charges to singularities of the
field. In electrostatics the two approaches, one based on the action at
distance and the other on an action mediated through the space, are equivalent.
In the study of quickly changing fields the usefulness of attributing a
physical reality to the electrical field appears evident, as it is possible to
create fields without the presence of charges.
The great
intuition of Maxwell was, in fact, the introduction of the concept of
displacement current (d). The importance of the
displacement current, which exists only in non-stationary cases, is in the fact
that it has the same properties as conduction currents. This means that a
quickly changing electrical field is surrounded by a magnetic field exactly as
a conductor in which current flows.
In this way,
the total current, i.e. conduction + displacement current (i + ∂d/∂t), has always
divergence zero. In other words, the total current flows in a closed loop, even
when there is no metallic or even no physical continuity in the circuit. In
metallic conductors, the conduction current is much higher than the
displacement current. This holds true until the fields become very quickly
changing. For example, if an electrical field in the vacuum has a value of 1000
V/m and a frequency of 10-8 sec., a surface 1 cm2 wide
would be crossed by a displacement current of about 10µA. In
insulating materials, vice versa, only the displacement current is present.
Not only electrostatic
and magnetostatic quantities, but also dynamic quantities and the generation
and propagation of electromagnetic waves find place in the same coherent theoretical
frame represented by Maxwell’s equations and nothing else needs to be added to
explain all these phenomena.
…or maybe not?
Not exactly.
The dissymmetry in the reciprocal action between a magnet and a conductor,
together with the negative attempts to focus the properties of the cosmic
ether, led another immense genius, Albert Einstein, to develop the theory of
relativity.
With his “zur
Elektrodynamik der bewegter Körper” the concept of cosmic ether, still an
open question at the time of Maxwell, became superfluous, because it was not
anymore necessary to conceive a stationary space with special properties. However,
Maxwell’s equations continued to hold true in the “classical” world, while their
tensorial covariant form ensured validity in any frame of reference compliant
with Lorentz transforms, clearly showing that electromagnetic phenomena do not
separately occur in space and time, but in the four-dimensional pseudo-Euclidean
space-time, where cinematics becomes geometry.
The
electrical field and the magnetic field, classically distinct, merge into a
single field: the field of the double emisymmetrical tensor, constituting the
electromagnetic tensor.With the theory
of relativity, new huge horizons were opened for the electromagnetism and not
only for it, creating the background for the bursting development of science
and technology of our times.
But this is another story.
But this is another story.
