Page 25 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 5

Source Denotation
(, ) Electrical current

(, ) Magnetic charge (predicted but not found yet)

(, ) Magnetic current (predicted but not found yet)

In what follows we will clarify the meaning of all these quantities. However, what can be said
about the nature of vector fields and their sources if they are pretty hard to visualize? We are
going to build the theoretical and quantitative description by observing and measuring their
interactions with surrounding physical objects. If such explanation of the phenomenon is quite
accurate and can be presented in the analytical or numerical form nothing prevents from
applying this knowledge to our practical problems hoping to get from physicists more profound
explanation of their nature later.

1.2 FUNDAMENTAL PRINCIPLES OF ELECTRODYNAMICS

1.2.1 Symmetry in Nature and Conservation Laws
The fundamental laws of all science and engineering disciplines are the conservation laws of
electric charge, mass, electromagnetic energy, momentum, etc. In physics, a conservation law
states that a particular measurable property of an isolated physical system does not change as
the system evolves over time. Intuitively we believe in it, and our practical life experience
convinces us to rely on it. The conservation laws are the most trusted and valuable source of
information about the complex, sometimes poorly understood, systems.

In 1915 the German mathematician Emmy Noether discovered the deep connection between
the symmetries of nature and conservation laws and “…showed on very general mathematical
ground that for physical theories of a particular type, every symmetry leads to a corresponding
conservation law.” [2] For example, if you place a set of charges in free space, far from anything
that might affect them (isolated system), it does not make a difference where exactly you put
the charges. There are no preferred sites in free space; all locations are equivalent. That
translation in space symmetry leads to the law of conservation of linear momentum, i.e. the
total product of mass and vector velocity is always conserved. Furthermore, it does not make a
difference at what time you start an experiment with the same system of charges in free space.
The results will be the same. That symmetry with respect to time shift (translation in time) leads
to the law of conservation of energy, maybe the most important conservation law in physics.

1.2.2 Conservation Laws
It is possible to prove (it is beyond the scope of this book) that Maxwell’s equations possess all
the required symmetries noted above. They belong to the group of physical theories that follow
fundamental laws of electrodynamics and can be considered as axioms [3]:

1. Electric charge conservation is linked to the symmetry of scale transformation or gauge.
Maxwell’s equations for point-to-point field description are a set of partial differential
equations. Since the derivatives of constants are equal to zero, we can add arbitrary
constants to their solutions. The actual situation is much more complicated. It can be shown
that we have “gauge freedom” or can insert the whole set of functions to the same solution!
Consequently, deriving a unique solution of Maxwell’s equations is not possible without
some additional conditions, and there is, therefore, no way to compare such an uncertain
solutions with indeed unique ElectroMagnetic (EM) fields measured in the real world. In