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Physics 3 - Maxwell
Note from the Author
Table of Contents
What is this book
Historical facts
New aspects
Fully erroneous
Incorrectly interpreted

Physics 3 - Chapter 1
Equations
Complex vectors form
The Most General form
The General Solution

Physics 3 - Chapter 2
Solutions
Initial conditions
Non-homogeneous equation
Solution for three-directions
The four laws

Physics 3 - Supplement
Fermat's proof
Beal's conjecture
Pythagorean triples
Inertial mass
Gravity constans big G
What does the Moon look at?

Physics 3 - Final notes
Final notes

Physics 4 - New book
Entry

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1.4.  Application of complex notation allows for
          the most general form of equations

     Since we already know the principle and reasoning of applying complex variables
to fields E and B we can apply a and b coefficients from formulas (1.23) and (1.24)
to all equations (1.6 – 1.9).
We assume the coefficients a = 1 c   and  b =i  which results for sum of (1.6) and (1.7) in: ∇ ⋅ ( 1 c ⋅ E ⃗ + i ⋅ B ⃗ ) = 1 c ⋅ ϵ 0 ⋅ ρ E + i ⋅ ρ M .
For (1.8) i (1.9), we divide first (1.9) by c2
   ∇ × B ⃗ = 1 ϵ 0 ⋅ c 2 ⋅ J E ⃗ + 1 c 2 ⋅ ∂ E ⃗ ∂ t . (1.27)
The sum of (1.8) with coefficient a = 1 c and (1.27) with coefficient i is equal to:
   ∇ × ( 1 c ⋅ E ⃗ + i ⋅ B ⃗ ) = ( i 2 c ⋅ J ⃗ M + i ϵ 0 ⋅ c 2 ⋅ J ⃗ E + i c 2 ⋅ ∂ E ⃗ ∂ t + i 2 c ⋅ ∂ B ⃗ ∂ t ) ,
therefore
   ∇ × ( 1 c ⋅ E ⃗ + i ⋅ B ⃗ ) = i c ⋅ ( i ⋅ J ⃗ M + 1 ϵ 0 ⋅ c ⋅ J ⃗ E + ∂ ∂ t ( 1 c ⋅ E ⃗ + i ⋅ B ⃗ ) ) . (1.28)
     We can see that by using the same coefficients a = 1 c and b =i , which were used
in (1.24) for P ⃗ = E ⃗ c + i ⋅ B ⃗ , following complex variables can be created for general sources:
   ρ = ρ E c ⋅ ϵ 0 + i ⋅ ρ M %rho = %rho sub E over {c cdot %epsilon sub 0} + i cdot %rho sub M , (1.29)

   J t ⃗ = J ⃗ E c ⋅ ϵ 0 + i ⋅ J ⃗ M vec J sub t = vec J sub E over {c cdot %epsilon sub 0} +i cdot vec J sub M 		(1.30)
and consequently, we can express Maxwell’s equations in shorter form:
   ∇ ⋅ P ⃗ = ρ nabla cdot vec P = %rho , (1.31)
   ∇ × P ⃗ = i c ⋅ ( J ⃗ t + ∂ P ⃗ ∂ t ) . (1.32)
     We have assigned the subscript t to variable J ⃗t thus designating variable J ⃗
without a subscript for even more general expression by introducing time variable τ= c⋅t, therefore (1.32) can be written as:
   ∇ × P ⃗ = i ⋅ ( J ⃗ + ∂ P ⃗ ∂ τ ) nabla times vec P = i cdot ( vec J +{{partial vec P} over {partial %tau}} )   , where J ⃗ = 1 c ⋅ J t ⃗ vec J = {1 over c} cdot vec J sub t (1.33)
This equation together with (1.31) present the shortest form of Maxwell’s equations and reduces sizes of resulting equations in subsequent operations. .

1.5.  Even More General Notation of Equations

      It seems like, there is no simpler notation of Maxwell’s equation than shown in (1.33). Perhaps simpler notation does not exist but further generalization is possible due to two possible versions of equations (similarly to 1.25 and 1.26) for two (left or right handed) coordinate systems.
Therefore, equation below
   ∇ × P ⃗ = − i ⋅ ( J ⃗ + ∂ P ⃗ ∂ τ ) nabla times vec P =- i cdot ( vec J +{{partial vec P} over {partial %tau}} ) (1.34)
represents Maxwell’s equations as well, so both versions, (1.33) and (1.34) should be always presented.
      However, both versions can be written in one equation when we square both sides of equations which yields:    ( ∇ × P ⃗ ) 2 = − ( J ⃗ + ∂ P ⃗ ∂ τ ) 2 ( nabla times vec P) sup 2 =- ( vec J +{{partial vec P} over {partial %tau}} ) sup 2 , therefore
   ( ∇ × P ⃗ ) 2 + ( J ⃗ + ∂ P ⃗ ∂ τ ) 2 = 0 ( nabla times vec P) sup 2 + ( vec J +{{partial vec P} over {partial %tau}} ) sup 2 =0 (1.35)
      This is interesting expression, and perhaps can inspire some philosophical reflections on the essence of Maxwell’s equations.
     It can be noted that ( ∇ × P ⃗ ) 2 ( nabla times vec P) sup 2 represents static, while   ( J ⃗ + ∂ P ⃗ ∂ τ ) 2 ( vec J +{{partial vec P} over {partial %tau}} ) sup 2 dynamic phenomena.
      Expression (1.35) includes both possible versions (1.33) and (1.34) thus represents the highest degree of generalization.
     It should be remembered that variable P ⃗  for (1.33) is expressed by   P ⃗ = E ⃗ c + i ⋅ B ⃗ , while for (1.34) is expressed by    P ⃗ = E ⃗ c − i ⋅ B ⃗ .
      In search for solutions of Maxwell’s equations it is sufficient to consider only one of the versions. Therefore, in later chapters we will deal only with version (1.33).

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