PPT No. 37
Development of Maxwell’s Equations
The laws /observations about Electricity and Magnetism
based on experiments by scientists like Coulomb, Gauss,
Ampère, Faraday were available in Maxwell’s time.
Maxwell expressed them in concise mathematical form.
In addition he contributed to their synthesis.
He thought that if changing magnetic field produces
electromotive force (according to Faraday’s law) then,
in view of the symmetry of physical laws,
a changing electric field should produce a magnetic field.
Development of Maxwell’s Equations
Maxwell derived a set of equations
that completely unified
the concepts of electric and magnetic fields
into one mathematical model.
Maxwell himself formulated his theory
in terms of twenty scalar equations.
Oliver Heaviside in England, considered as
one of the fathers of vector analysis,
abstracted Maxwell’s set of equations
to four equations in the vector form.
Development of Maxwell’s Equations
The four equations, abstracted by Oliver Heaviside
are fairly elaborate, vector equations &
involve abstract mathematical terminology
as curl and divergence
They are called as “Maxwell's equations”
since A. Einstein called them as “Maxwell's equations”
They can be presented in two sets
1) Microscopic Equations
2) Macroscopic Equations as follows
Maxwell's Microscopic Equations
Microscopic equations are the most basic ones and
describe the electric field E and the magnetic field B
in vacuum, together with their sources
(charge-density and current-density)
ρ is the position dependent density of electric charge,
J is the position dependent density of electric current,
μo and εo are the magnetic and electric constants
Maxwell's Microscopic Equations-Integral form
in the absence of Magnetic or Polarizable media:
1. Gauss' law for electricity
2. Gauss' law for magnetism
3. Faraday's law of induction
4. Ampere's law
Maxwell's Microscopic Equations-Differential form
In the absence of magnetic or polarizable media
1. Gauss' law for electricity
2. Gauss' law for magnetism
3. Faraday's law of induction
4. Ampere's law
∇ x B = μoJ + μoεo∂E/∂t
Maxwell's Macroscopic Equations
Macroscopic equations are applicable
(instead of the vacuum) to
a continuous medium (e.g. air)
and a medium that is polarizable and magnetizable.
In material media
that are responsive to electric and magnetic fields,
charges and currents are set up in the presence of fields.
The induced fields
either tend to cancel or augment the applied fields,
depending on the properties of the material.
Then Maxwell’s equations contain different notation
Maxwell's Macroscopic Equations
In material media
it is more convenient to use the auxiliary fields
since they can be calculated using
only the free charges and currents.
From the auxiliary fields
the actual fields can be calculated
considering the specific properties of the responses.
Maxwell's Macroscopic Equations
In particular, in this formulation,
the auxiliary fields are defined in SI units
by the constitutive relations
E = (D-P)/εo,
where P is the polarization per unit volume,
and is determined by the bound charge
B = (H+M)*μo,
where M is the magnetization and
is determined by the bound current
Maxwell's Macroscopic Equations-Differential form
with magnetic and/or polarizable media
1. Gauss' law for electricity
2. Gauss' law for magnetism
3. Faraday's law of induction
4. Ampere's law
Comments on Maxwell’s Equations
George Galeczki in his article entitled
“Beyond Maxwell-Lorentz Electrodynamics”,
has commented on Maxwell’s Equations (ME) as follows
1)The basic formulation of ME –
as derived from experiments –
is in integral form pertaining to
a finite, closed area or volume.
2) The ME are formulated for continuous fields and
are called, therefore, field equations.
3) ME hold for closed circuits only
Comments on Maxwell’s Equations
4) The sources of the fields are
continuous charges and continuous current densities.
The discrete, quantized charges
introduced in Maxwell's theory are foreign elements.
5) ME are tautological in the sense that
they merely represent relationships
between fields and their sources.
One has to provide the initial charge distribution
in order to be able to calculate the field distribution,
or vice versa.
Comments on Maxwell’s Equations
6) ME are unable to describe the interaction
between two discrete charges.
7) ME are unable to supply the equation of motion
of one charge in the field produced by all others.
8)ME are not suited for the description of
open circuits like antennas.
Comments on Maxwell’s Equations
9) ME are unable to prescribe
the exact conditions under which
a system will radiate, or not.
The notorious example is
Bohr's planetary model of the hydrogen atom.
10) ME are unable to provide
a stable model for the elementary charge.
Comments on Maxwell’s Equations
11) ME are generally covariant and
do not single out the Lorentz transformation (LT)
of the "special" theory of relativity (STR).
12) ME are formulated in terms of
independent, Eulerian coordinates x, y, z, t and
partial derivatives