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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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 I.5. Incorrectly interpreted specific problems

        The good example of incorrect interpretation is an issue (described in numerous textbooks) regarding presentation of spherical wave equation.
Presenting this equation in polar coordinates with assumption that value of F ⃗ ( r ⃗ , τ ) vec F( vec r,%tau) wzori2a, depends only on r and τ allows to use the relation:

   ∇ 2 F ⃗ ( r ⃗ , τ ) = 1 r ∂ 2 ∂ r 2 ( r ⋅ F ( r , τ ) ) nabla sup 2 vec F( vec r,%tau)= {1 over r}{{partial sup 2 } over {partial r sup 2}} ( r cdot F( r,%tau)) wzori1a, (i.1)

which results in the following wave equation:

   1 r ∂ 2 ∂ r 2 ( r ⋅ F ( r , τ ) ) − 1 r ∂ 2 ∂ τ 2 ( r ⋅ F ( r , τ ) ) = 0 {1 over r}{{partial sup 2 } over {partial r sup 2}} ( r cdot F( r,%tau)) - {1 over r}{{partial sup 2 } over {partial %tau sup 2}} ( r cdot F( r,%tau)) = 0 wzori2a, (i.2)

which is subsequently substituted by:

   ∂ 2 ∂ r 2 ( r ⋅ F ( r , τ ) ) − ∂ 2 ∂ τ 2 ( r ⋅ F ( r , τ ) ) = 0 {{partial sup 2 } over {partial r sup 2}} ( r cdot F( r,%tau)) - {{partial sup 2 } over {partial %tau sup 2}} ( r cdot F( r,%tau)) = 0 , (i.3)

so, we can derive mathematics solution:

   F ( r , τ ) = 1 r ⋅ ( ϕ ( r + τ ) − ψ ( r − τ ) ) F( r,%tau)= {1 over r} cdot( %phi ( r+%tau)-%psi( r-%tau)) , (i.4)

         The above equation is however useless in application for physics since value on
the right side represents zero source (no source) i.e. there is practically no wave.
The correct wave equation (i.2) should be then written as:

   1 r ∂ 2 ∂ r 2 ( r ⋅ F ( r , τ ) ) − 1 r ∂ 2 ∂ τ 2 ( r ⋅ F ( r , τ ) ) = f ( r = 0, τ ) {1 over r}{{partial sup 2 } over {partial r sup 2}} ( r cdot F( r,%tau)) - {1 over r}{{partial sup 2 } over {partial %tau sup 2}} ( r cdot F( r,%tau)) = f( r=0, %tau ) , (i.5)

then (i.3) becomes:

   ∂ 2 ∂ r 2 ( r ⋅ F ( r , τ ) ) − ∂ 2 ∂ τ 2 ( r ⋅ F ( r , τ ) ) = r ⋅ f ( r = 0, τ ) {{partial sup 2 } over {partial r sup 2}} ( r cdot F( r,%tau)) - {{partial sup 2 } over {partial %tau sup 2}} ( r cdot F( r,%tau)) = r cdot f( r=0, %tau ) , (i.6)

The solution of this (i.6) equation is not as simple as solution of previous (i.3) equation. We can conclude there is singularity in the point where r = 0.

        The problem presented above was pointed by Feynman
(textbook [L.1] – vol. II, paragraph 20-4 Spherical waves),
which contains the following passage:
“We must mention another important point. In our solution for an outgoing wave, Eq.(20.35), the function ψ is infinite at the origin. That is somewhat peculiar. We would like to have a wave solution which is smooth everywhere. Our solution must represent physically a situation in which there is some source at the origin. In other words, we have inadvertently made a mistake. We have not solved the free wave equation (20.33) everywhere; we have solved Eq.(20.33) with zero on the right everywhere, except at the origin. Our mistake crept in because some of the steps in our derivation are not “legal” when r=0.”
        Similar problem is commented in [L1] (paragraph 21 – Solution of Maxwell’s equations with Currents and Charges) where we read:
“It turns out that we won’t quite make it—that the mathematical details get too complicated for us to carry through in all their gory details.”

         The problem of incorrect interpretation results from wrong approach – instead of thoroughly solving differential equations, the solutions are guessed. When some abnormalities manifest, then one looks for some justifications.

Paragraph (2.2.3) in this book presents correct interpretation of dependency on square of distance which eliminates problems described above.


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