axiomatic reformulation of maxwell

8/22/2019 Axiomatic Reformulation of Maxwell

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Axiomatic Reformulation of Maxwells Theory of Electromagnetism

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Axiomatic Reformulation of MaxwellsTheory of ElectromagnetismR. ANNOU

(1989/1990)

. 1. IntroductionThe laws of electricity and magnetism were developed on the basis of observations and

experimental facts, and were not derived from conservation principles in the framework of a

general theory. Indeed, the electrical effect was observed during the eighteenth century (1700)

by some pioneers such as Lord Kelvin, Coulomb, Volta, Cavendish, etc., by way of numerous

experiments. It has been observed that matter possesses the ability of attracting shreds of a sheet

of paper after rubbing that matter (a rod of glass or resin) against the skin. This characteristic

quality is represented by a physical quantity called--- the electric charge. Similarly, mass is

representing the attribute of inertia. On the other hand, the magnetic effect was already known.

However, the relation between the electric current and magnetism was not established until

discovered by H.C. Oersted in 1820. Later on, A. Ampre, M. Faraday, and J. Henry studied

thoroughly the magnetic effect.

Finally, J. C. Maxwell found the missing piece in the puzzle, namely, the displacement

current and succeeded in building a self-consistent, compact and powerful theory ofelectromagnetism. In this short note we shall prove that Maxwell equations can be deduced

from an axiomatic based on simple and general axioms.

. 2. AxiomaticFirst axiom

The electric charge (Q) is conserved.

Second axiomThe electric charge (Q) acts on the space and influences other charges put in this space.

Third axiomThe energy is conserved.

. 3. Mathematical interpretation of the axiomatic1/ Let be a tube; the first axiom means that the amount of electric charge getting out of the tube

is equal to the amount of electric charge getting in. Let be the current density vector given by,


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, where, is the charge density and is the charge velocity. Hence, the chargeconservation may be expressed through the following equation,

(1)

2/ a. The charge acts on the space, means that the charge exhibits a certain entity that can be

defined in all the space. This influence on the space is directional so one may choose that entity

to be a vectorial field , the flux of which through a bounded surface surrounding the charge Q,measures the later. Therefore, the mathematical formulation of the axiom is given by,

(2)

where, is a unit-system dependent constant. It is in cgs system.

The local form is given by,

(3)

b. Influences other charges put in this space means, ifwe put a test charge q in the space of

Q, the latter will influence the movement of the former by a force , which depends on q andQ, consequently on q and . It is written then as follows,

The simplest form would be linear,

It is to be stressed, that the above equation has been deduced by way of simple arguments.

3/ In a definite volume V, the variation of the field energy is due to two facts,

the field yields energy to matter,

or,

the field loses energy by radiation through the surface enclosing volume V.

Let u be the energy density and the flux of energy. So the loss of energy is given by,

(work given to matter per time unit), (4)

The work per unit time that is power is given for N particles per unit volume by Power = . Hence, the energy equation may be rewritten as,

( )

(5)


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Or in the local form,

, (6)

Finally, we got the following set of equations,

From Eqs. (1) and (3), we get,

(7)

Hence it does exist a vector the curl of which is given by,

(8)

We may choose to write,

to comply with the notations already used for the magnetic field in the electromagnetism theory.

Hence the existence of a physical quantity called the electric charge gives ultimately rise to what

we know as magnetism.

It is to be noticed also that the displacement current appears automatically in the

equation, as a logical consequence of the above mentioned axioms. We need not to invoke any

argument to introduce it, the way it is being done so far.

Let us implement the axiomatic by proposing a form to and . The electric charge Qinfluences the surrounding through the fields

and

, hence,

and,

The simplest form would be,

(9)

and,

(10)


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By using the identity, , one finds after putting Eqs. (9,10) inEq. (6) the following relation,

[ ( ) ] (11)

where, we chose to retrieve Maxwell equations. It should be kept in mind that the setof solutions of the equations derived by the following approach is wider as the solutions of

Maxwell equations constitute only a sub-set of it; especially that we consider only the solutions

satisfying the following equation,

( ) (12)

or,

( ) (13)

By applying the divergence (multiply by the gradient) on the two sides of Eq.(13), one finds,

( ) (14)

Equation (14) may be rewritten consequently as,

(15)

has the form of a magnetic charge density corresponding to magnetic monopoles. It is

important to notice that the present approach does not reject the existence of magnetic

monopoles, in which case the current density vector ought to be conservative ( .However, if we want to follow the track of Maxwells theory of electromagnetism we need to

recall that it should exist a time (in the past) where vanishes, and then,

. (16)

. 4. ConclusionTo conclude, we recall that by way of general considerations (basic principles), namely, the

conservation of charge and energy, we could derive Maxwells equations,

( )

axiomatic reformulation of maxwell

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