The Imagineer's Chronicles:
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Chapter Twenty-Four

Maxwell's equations
in terms of Four *spatial* Dimensions

Maxwell's equations can be derived in terms laws of classical physics, the existence of four *spatial* dimensions and a continuous non-quantized form of mass.

Chapter two derived the particle properties of mass, energy and electromagnetic radiation in terms of the integral energies associated with a resonant "system" formed in space by "oscillations" in a continuous non-quantized form of mass.

Chapter three defined the propagation of electromagnetic energy in terms of matter wave in a continuous non-quantized form of mass  moving at the velocity of light on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Therefore, Chapters two and three define a common mechanism responsible for particle/wave duality of electromagnetic energy in terms of the classical laws of physics, the existence of four *spatial* dimensions and a continuous non-quantized form of mass.

(Louis de Broglie was the first to theorize that all particles had a wave component.  His theories were confirmed by the discovery of electron diffraction by crystals in 1927 by Davisson and Germer.  However, this means there must be a continuous non-quantized medium for it to be propagated on because even the smallest possible particle must have a wave component.  However, macroscopic observations of wave energy indicate that it can only be propagated on a medium made up of mass.  Therefore, the success of Louis de Broglie theory indicates that a continuous non-quantized form of mass exists.)

However, the existence of a continuous non-quantized form of mass and four *spatial* dimensions also provides the ability to define a mechanism responsible for Maxwell's equations in terms of the classical laws of physics

The "oscillations" in a continuous non-quantized form of mass responsible for generating a matter wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would be analogous to the oscillations associated with a wave on water.

In a classical world, the trough of a wave on water displaces the vertical volume of water below the surface of the water with air while its peak displaces the air above the surface of the water with water.

Similarly, a "trough" of a matter wave on a "surface" of a three-dimensional space manifold would displace a volume of three-dimensional space below its "surface" with a component of four-dimensional space.  While a "peak" of a matter wave would displace a "volume" of four-dimensional space above a "surface" of a three-dimensional space manifold with a continuous non-quantized mass component of three-dimensional space.

Chapter ten derived all forms of energy in terms of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

As mentioned earlier Chapter three defined the propagation of electromagnetic energy in terms of matter wave in a continuous non-quantized form of mass moving at the velocity of light on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This means the sinusoidal displacements in a "surface" of a three-dimensional space manifold caused by the "peaks" and "troughs" of a matter wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension would be responsible for the propagation of electromagnetic flux.

First the mechanism responsible for Gauss's law of Electricity, the total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity will be derived in terms of the geometric properties of four *spatial* dimensions.

Next Gauss's law of Magnetism, the net magnetic flux out of any closed surface is zero and that there are no magnetic monopoles will be defined in terms of the alignment of the axis of a fourth *spatial* dimension with respect to the axes of three-dimensional space.

Following that the physical mechanism responsible for Ampere's Law, the magnetic field in space around an electric current is proportional to the electric current that serves as its source, will be derived in terms of an energy gradient in a three-dimensional space manifold that Chapter thirteen derived as being responsible for an electric charge. 

Then the fourth of Maxwell's equations or Faraday's law of induction that any change in a magnetic environment of a coil will cause a voltage to be "induced" in the coil that is equal to the negative rate of change of the magnetic flux times the number of turns of the coil will be derived in terms an energy gradient associated with a charge and inertial properties of a continuous non-quantized form of mass defined in Chapter one.

Gauss's law of electricity states the total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity or "integral of E*dA =Q / permittivity".

The mechanism responsible for the total electric flux out of a closed surface being equal to the charge enclosed is related to the "peaks" and "troughs" of a matter wave on a "surface" of a three-dimensional space manifold.

Chapter nine derived the magnitude of an electric charge in terms of a displacement of the surfaces" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.  A positive charge was defined in terms of a "depression" in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension while a negative charge was defined in terms of an "elevation" in that same surface.

As mentioned earlier the displacements in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by the "peaks" or "elevations" and "troughs" or "depressions" of a matter wave are responsible for the propagation of electrical flux through space.

However, as was shown in Chapter two and three that all particle are composed of a matter wave because they shown that a particle of energy is made up of a standing matter wave on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

Therefore, as was shown in Chapter nine the displacements caused by the alternating "elevations" and "depressions" in a "surface" of a three-dimensional space manifold generated by this standing matter wave would be responsible for the transmission of positive and negative electrical flux through space.

Therefore, the total electric flux or energy out of a closed "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension surrounding a charge will be equal, as Gauss's Law of Electricity states, to number of charge particles enclosed by that surface.

The fact that the total electric flux out of a closed surface must be divided by the permittivity of the volume of space an electric flux is transversing is related to the mechanism responsible for the propagating electric flux in space.

As mentioned earlier electrical flux is propagated thought space by a matter wave in a continuous non-quantized form of mass moving at the velocity of light.

However, Chapter twelve showed there would be direct relationship between the density or permittivity of space and the density of a continuous non-quantized form of mass in that space. 

Therefore, the electric flux out of a closed surface must be divided by the permittivity of a volume because the attenuation of the energy of a standing matter wave in a continuous non-quantized form of mass would be directly related to the density of the continuous non-quantized form of mass that it was moving through.

This completes the derivation of the physical mechanism responsible for Gauss's law of electricity in terms of the existence of four *spatial* dimensions and a continuous non-quantized form of mass.

We will now derive the mechanism responsible for Gauss's law for magnetism, that the net magnetic flux out of any closed surface is zero and that magnetic monopoles do not exist or the integral of B*dA=0 in terms of the existence of four *spatial* dimensions.

The mechanism responsible for net magnetic flux out of a closed surface being equal to zero is related to the "alignment" or angle the axis of a fourth *spatial* dimension has with respect to coordinate planes of a three-dimensional space.

(The axis of the fourth *spatial* dimension was defined in Chapter ten.)

Chapter eleven showed magnetic flux was due a "tangential" force generated on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension by a "misalignment" of the axis of a fourth *spatial* dimension with respect to the coordinate plains of three-dimensional space.

Therefore, the misalignment of a "surface" of a three-dimensional space manifold with respect to the axis of a fourth *spatial* dimension caused by the "peaks" and "troughs" of a matter wave on a "surface" of a three-dimensional space manifold would be responsible for the alternating polarity of the magnetic flux associated with propagation of electromagnetic energy.

However the "angle" or "alignment" the axis of the fourth *spatial* dimension makes with a coordinate plain in a three-dimensional space manifold would be analogous to the angle a line makes when drawn through a plain on a coordinate graph.

A line will always make symmetrically equal and opposite angles above and below the coordinate plain it transverses.

Similarly, the axis of a fourth *spatial* dimension will always make symmetrically equal and opposite angles "above" and "below" the coordinate plain in three-dimensional space.

Therefore, as was indicated in Chapter eleven the net magnetic flux or force out of any closed surface always being equal to zero because the magnitude and "direction" of a force responsible for magnetic flux will be equal and opposite "above" and "below" a three-dimensional coordinate plain it is transversing.

This also defines the physical reason why magnetic monopoles cannot exist.

Magnetic monopoles cannot exist because the axis of a fourth *spatial* dimension is infinite in length

This is analogous to how the each axis in three-dimensional space is infinite in length.

Therefore, magnetic monopoles do not exist because the axis of a fourth *spatial* dimension is infinite in length it must generate two magnetic fields of equal and opposite polarity "above" and "below" any coordinate plain in three-dimensional space it transverses.

This completes the derivation of the physical mechanism responsible for Gauss's law of Magnetism in terms of the orientation of the axis of a fourth *spatial* dimension with respect to the axes of three-dimensional space.

We will now move on and derive a physical mechanism responsible for Ampere's law that the magnetic field in space around an electric current is proportional to the electric current which serves as its source or the integral of B*ds=the permittivity of space times the current or “I”.

The mechanism responsible for Ampere's law can be derived from the "alignment" of the axis of a fourth *spatial* dimension with respect to the tangential forces associated with an electron defined in Chapter thirteen.

Chapter thirteen derived charge in terms of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension while Chapter eleven defined the magnetic flux associated with a moving charge in terms of a tangential force on a "surface" of a three-dimension space manifold caused by that displacement.

In an electric current, the tangential forces associates with the magnetic flux of a charge will have a common "alignment" along the velocity vector associated with the electrically charged particles.

Therefore, the "tangential" force, which defines magnetic field in Chapter eleven, on a "surface" of a three-dimensional space manifold by the individual particles in an electric current will be aligned and additive because each will contribute  to the "tangential" force on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This results in the magnetic field in space around an electric current to be directly proportional to the electric current because, as was shown in Chapter eleven, the magnitude of a magnetic field is directly proportional to the magnitude of a "tangential" force along a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This completes the derivation of the mechanism responsible for Ampere's law in terms of the "alignment" of the "W" of the fourth *spatial* dimension and the existence of the existence of four *spatial* dimensions

Lastly we shall derive the mechanism responsible for Faraday's Law of Induction, that any change in a magnetic environment of a coil will cause a voltage to be "induced" in that coil that is equal to the negative rate of change of the magnetic flux times the number of turns of the coil or the integral of E*ds= -dB/dt.

The mechanism responsible for Faraday's Law of Induction is related to the inertial properties of a continuous non-quantized form of mass responsible for a unit charge defined in Chapter thirteen.

In the derivation of Gauss's law of Electricity it was shown why the total electric flux out of a closed surface is equal to the charge enclosed in terms of the existence of a continuous non-quantized form of mass and four *spatial* dimensions.

This means each additional turn of a coil will add charges to the space effected the movement of them through that coil.  This will increase the total "tangential" or force generated on a continuous non-quantized form of mass in the space affected by those charges.

Later in the derivation of Ampere's law, it was shown the "tangential" force generated on a continuous non-quantized form of mass by an electric current is directly related to the number of changed particle

This means each additional turn of a coil will make a proportional increase the magnitude of the magnetic flux or force associated with the current in that coil.

This defines the mechanism responsible for why the magnitude of the magnetic flux around that coil that is carrying an electric current is directly proportional to the number of turns in that coil. 

One aspect of electrical energy that has not yet been mentioned in this chapter is the mechanism responsible for the force or voltage associated with the unit electric charge.

Chapter nine derived the energy associated with the voltage and magnetic flux of an electric charge in terms of energy gradients, which as mentioned earlier, are caused by the "peaks" and "troughs" of a matter wave in a continuous non-quantized form of mass.

This suggests the change in the magnetic environment of a coil, which causes a voltage to be "induced" in that coil, may be related to the inertial properties of a continuous non-quantized form of mass associated with a charged particle.

This means that Newton’s third law of motion involving an interaction between masses which states "For every action there is an equal and opposite reaction" may be applied to the magnetic environment of a coil.

In other words, the voltage is induced in a coil is equal to the negative rate of change of the magnetic flux because a change in a continuous non-quantized component of space associated with a voltage could be considered to be an equal and opposite reaction to the change in a continuous non-quantized mass component space associated with a magnetic flux based on Newton's third law of motion.

This completes the derivation of the physical mechanism responsible for Faraday's law in terms of the inertial properties of a continuous non-quantized form of mass responsible for a unit charge defined in Chapter thirteen.

This also completes the derivation of Maxwell's equations in terms of the existence of four *spatial* dimensions, a continuous non-quantized form of mass and the laws of classical physics.

The universe's most powerful enabling tool is not
  knowledge or understanding but imagination
  because it extends the reality of
  one's environment.

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Copyright 1995 Jeffrey O'Callaghan