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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