Page 17 - Electromagnetics Handbook

stresses and strains in media surrounding charged objects. His law of induction was
formulated not in terms of positions of bodies, but in terms of lines of magnetic force.
Inspired by Faraday’s ideas, Gauss restated Coulomb’s law in terms of flux lines, and
Maxwell extended the idea to time changing fields through his concept of displacement
current.
In the 1860s Maxwell created what Einstein called “the most important invention
since Newton’s time”— a set of equations describing an entirely field-based theory of
electromagnetism. These equations do not model the forces acting between bodies, as do
Newton’s law of gravitation and Coulomb’s law, but rather describe only the dynamic,
time-evolving structure of the electromagnetic field. Thus bodies are not seen to inter-
act with each other, but rather with the (very real) electromagnetic field they create,
an interaction described by a supplementary equation (the Lorentz force law). To bet-
ter understand the interactions in terms of mechanical concepts, Maxwell also assigned
properties of stress and energy to the field.
Using constructs that we now call the electric and magnetic fields and potentials,
Maxwell synthesized all known electromagnetic laws and presented them as a system of
differential and algebraic equations. By the end of the nineteenth century, Hertz had
devised equations involving only the electric and magnetic fields, and had derived the
laws of circuit theory (Ohm’s law and Kirchoff’s laws) from the field expressions. His
experiments with high-frequency fields verified Maxwell’s predictions of the existence of
electromagnetic waves propagating at finite velocity, and helped solidify the link between
electromagnetism and optics. But one problem remained:if the electromagnetic fields
propagated by stresses and strains on a medium, how could they propagate through a
vacuum? A substance called the luminiferous aether, long thought to support the trans-
verse waves of light, was put to the task of carrying the vibrations of the electromagnetic
field as well. However, the pivotal experiments of Michelson and Morely showed that the
aether was fictitious, and the physical existence of the field was firmly established.
The essence of the field concept can be conveyed through a simple thought experiment.
Consider two stationary charged particles in free space. Since the charges are stationary,
we know that (1) another force is present to balance the Coulomb force between the
charges, and (2) the momentum and kinetic energy of the system are zero. Now suppose
one charge is quickly moved and returned to rest at its original position. Action at a
distance would require the second charge to react immediately (Newton’s third law),
but by Hertz’s experiments it does not. There appears to be no change in energy of
the system:both particles are again at rest in their original positions. However, after a
time (given by the distance between the charges divided by the speed of light) we find
that the second charge does experience a change in electrical force and begins to move
away from its state of equilibrium. But by doing so it has gained net kinetic energy
and momentum, and the energy and momentum of the system seem larger than at the
start. This can only be reconciled through field theory. If we regard the field as a
physical entity, then the nonzero work required to initiate the motion of the first charge
and return it to its initial state can be seen as increasing the energy of the field. A
disturbance propagates at finite speed and, upon reaching the second charge, transfers
energy into kinetic energy of the charge. Upon its acceleration this charge also sends out
a wave of field disturbance, carrying energy with it, eventually reaching the first charge
and creating a second reaction. At any given time, the net energy and momentum of the
system, composed of both the bodies and the field, remain constant. We thus come to
regard the electromagnetic field as a true physical entity:an entity capable of carrying
energy and momentum.

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