Originally posted by Rakarskiy
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https://en.wikipedia.org/wiki/Maxwell%27s_equations
The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.
Maxwell's addition to Ampère's law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space.
Maxwell's addition to Ampère's law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space.
Both of these are incorrect, because they are in violation of the fundamental theorem of vector calculus aka the Helmholtz decomposition, which establishes a fundamental separation of linear (longitudinal) and angular (rotational) fields. The linear half of the Helmholtz decomposition is curl free and compressible, while the angular half is divergence free and incompressible.
The linear half involves the divergence operator over a closed volume resulting in a scalar potential, while the angular half involves the curl operator over a closed surface resulting in a vector potential, since rotation always goes around an axis which has a direction. Anyone who maintains that the electric field should have a curl and that a "displacement current" should be included in Ampere's law simply has not understood the definitions of these operators nor the (spatial) consequences thereof. A closed volume is simply not the same thing as a closed surface and that's ultimately why one cannot make the additions Maxwell did!
What Maxwell did is like having Pythagoras theorem:
a^2 + b^c = c^2
and Maxwell writes:
a^2 + b^c = c^2 + db/dt.
One simply cannot do things like that!
The proper relation between space and time is much more fundamental than on the one hand forcing the curl free half of the Helmholtz decomposition, the electric field, to have a curl and on the other hand to add (the time derivative of) a curl free term to the curl of the magnetic field. What I discovered is that the time derivative of any given vector field is given by combining the quantum circulation constant k with the vector Laplacian:
dF/dt = -k Delta F,
with Delta the vector Laplacian, *the* second spatial derivative in three dimensions.
And as I wrote in my (very preliminary) article:
https://github.com/l4m4re/notebooks/..._physics.ipynb
Obviously, the units of measurements match: [/s] = [m^2/s] [/m^2].
This is very significant, because it enables us to define higher order Laplace and Poisson equations in three dimensions and it also gives us the opportunity to obtain a much deeper understanding of the Nature of space and time in our Universe.
In other words: **this is the fundamental equation that will one day be recognized as one of the biggest scientific breakthroughs of the 21st century**, because there is simply no argument to be made against such a simple and straightforward application of the vector Laplacian.
This is very significant, because it enables us to define higher order Laplace and Poisson equations in three dimensions and it also gives us the opportunity to obtain a much deeper understanding of the Nature of space and time in our Universe.
In other words: **this is the fundamental equation that will one day be recognized as one of the biggest scientific breakthroughs of the 21st century**, because there is simply no argument to be made against such a simple and straightforward application of the vector Laplacian.
a = -k Delta v,
and the jerk field j by:
j = -k Delta a,
to obtain second order fields and then the (full 3D vector) wave equation becomes:
a/k + j/c^2 = 0,
indicating that the relation between the fields is actually much more complex as what Maxwell wrote and involves second order fields, of which current density J is related to the second order vector potential representing turbulence, while the second order scalar potential is related to temperature.
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