Symmetry of Maxwells Equations

1
The Early Days of Electromagnetism
By Benjamin Redshaw
Symmetry of Maxwells Equations Following
Maxwells death, Heaviside developed the idea of
vector notation and used this concept to
successfully transform the Maxwells original
equations into the four fundamental field
equations which are the ones that are common
place today. Following the reformulation of
Maxwells equations by Heaviside, it can be
clearly seen that there is an asymmetry in the
equations. Maxwells equations represented in
vector form consist of two homogeneous and two
inhomogeneous equations, namely Furthermo
re the reason for this asymmetry is that there
exist isolated point electric charges, but not
magnetic charges (an experimental fact). Oliver
Heaviside developed a version of electromagnetic
theory that was completely symmetrical between
electricity and magnetism. This was achieved by
introducing magnetic charges such that the
charges of the north and south poles would be of
different sign. Including the existence of
magnetic monopoles, Maxwells equations will take
the symmetrical form of These equations
would be invariant under the following
transformations And this is called a
duality transformation.
Electromagnetism Pre-Maxwell The extensive
study of electricity and magnetism began with the
research of the French physicist Charles Augustin
Coulomb who in 1787 proposed a law of force for
charges that varied inversely as the square of
the distance. Through experiments he used a
sensitive torsion balance demonstrating its
validity for forces of both repulsion and
attraction. Like Newtons law of gravitation,
Coulombs law was based on the concept of "action
at a distance," wherein bodies can interact
instantaneously and directly with one another
without the intervention of any intermediary.
Hans Christian Oersted made the momentous
discovery that an electric current in a wire
could deflect a magnetic needle. Following this
discovery, the laws of force between current
bearing wires were investigated by Andre-Marie
Ampere, Jean-Baptiste Biot and Felix Savart and
within six years the theory of steady currents
was complete. These laws were also "action at a
distance" laws, in other words they were
expressed directly in terms of the distances
between the current elements. In 1831, Michael
Faraday demonstrated the reciprocal effect, in
which a moving magnet in the vicinity of a coil
of wire produced an electric current. This,
together with Oersteds experiment with the
magnetic needle, led Faraday to conceive the
notion of a magnetic field. A field produced by a
current in a wire interacted with a magnet. Also,
according to his law of induction, a time varying
magnetic field incident on a wire would induce a
voltage, thereby creating a current. Now electric
forces could similarly be expressed in terms of
an electric field created by the presence of a
charge. Faradays field concept implied that
charges and currents interacted directly and
locally with the electromagnetic field, which
although produced by charges and currents, had an
identity of its own. Faraday was a self-taught
experimentalist and did not formulate his laws
mathematically resulting in his new concept
contrasting the idea of "action at a distance,"
which assumed bodies interacted directly with one
another.
References Jackson, J.D (1999), Classical
Electrodynamics, John Wiley Sons, Inc. Dirac,
P.A.M (1978), Directions in Physics, John Wiley
Sons, Inc. Pais, A, Jacob, M, Olive, D.I and
Atiyah, M.F (1998), Paul Dirac The Man and His
Work, Cambridge University Press Salam, A and
Wigner, E.P (Eds) (1972), Aspects of Quantum
Theory, Cambridge University Press
Project Supervisor Professor Patrick Dorey