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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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Chapter 2. Maxwell’s Equations Solution 2.1. Proper solution of Maxwell’s equations The method of solving equations (1.44) or (1.45) is identical, so we can deal with solving equations in their general form:
,	(2.1)
where and take only real values. Equation (2.1) represents concise version of six independent equations which consist of three equations for E⃗ , vector: Ex, Ey, Ez and three equations for B⃗ , vector: Bx, By, Bz. It has to be noted that in our case equations for E⃗ and B⃗ are independent of each other, and each contains on the right-side respective source. This feature also applies for individual vector components, so for further considerations we can treat them as independent scalar functions. They are expressed in similar form of wave equation:
	(2.2)
Subscript index „s” replaces indices „x”, „y” and „z” for individual vector components. Caution: Do not confuse directions of components of vectors F⃗ and f⃗ with directions of components of position vector r⃗ . 2.2. One-direction solution 2.2.1. Solution for homogenous equation In order to understand better principles of wave formation and its properties, we will begin with homogenous equation:
.	(2.3)
For better clarity of our equations, we don’t need to distinguish individual components of vectors (subscript index “s”) since individual calculations for them are the same. To solve equation (2.3) let us introduce auxiliary variables u and v as shown below:
u = x + τ , v = x - τ ,	(2.4)
which are bijective linear mapping functions.
,	(2.5)
Practically this represents rotation of coordinate system by π/2 along with scaling change. The rotation of coordinates without changing of scale can also be exercised but it does not gain any advantage, just opposite – equations become longer. As a result from variable substitutions (per 2.4) we obtain the same values of functions, but their forms are different. To distinguish the functions after substitution of x , τ with u ,v Zamieniając zmienne zgodnie z zależnościami (2.4) otrzymujemy te same wartości funkcji, ale ich postacie już różnią się i dla ich rozróżnienia funkcję, po zamianie parametrów x , τ na u ,v we will rename them to Fo(). (“o” comes from Polish word “obrot” meaning rotation) It is correct that F(x, τ) = Fo(u, v) so we can used them interchangeably: , and , . In our case notation of function F(x, τ) means that first variable relates to position, second variable to time, but variable names should not be associated with physical quantities since notation F(τ, x) would mean that time variable was replaced by position variable. In our case this can occur since both variables x and τ have the same dimensions. The same applies in our case for function Fo(u, v), where first variable relates to position in regard to u axis, and second variable position in regard to v axis. Notation Fo(v, u), would indicate putting variable v in place of u axis and putting variable u in place of v axis, which can occur in our conversions of equations. Using new variables, we obtain:
.	(2.6)
and respectively,
.	(2.7)
.	(2.8)
.	(2.9)
and finally:
.	(2.10)
In order that , derivative in respect to one variable cannot be dependent on second variable, which occurs when Fo() represents a sum of any two functions; let’s call them Φ() and Ψ() and each function depends only on single variable, either u or v, therefore:
Fo(u, v) = Φ(u) + Ψ(v) = Φ(x + τ) + Ψ(x - τ)	(2.11)
The reasoning above may create an impression that we are trying to guess some particular solution. Therefore, this will be formally derived. We have then:
.	(2.12)
Depending on an order of differentiating (integrating), we obtain: and , which means , represents derivative of some function, let’s call it Φ(), which depends only on variable u, and represents derivative of some other function, let’s call it Ψ() which depends only on variable v, therefore,
oraz	(2.13)
.	(2.14)
This means that both functions Φ(u) and Ψ(v) , and thus their sum (2.11) represent solution of equation (2.3). We could then conclude that the problem was solved, but equation (2.11) indicates only that solution exists, and functions Φ(u) and Ψ(v) represent any functions in spaces where equations are nonhomogeneous, i.e. right side of equation (2.1) is different than zero. All we can say about solutions of homogenous equation (2.3) is only that these are two waves moving in opposite directions Φ(x + τ) and Ψ(x - τ). We can also point out some interesting features of these functions. We will use also corresponding to them two-argument functions Φ_t (x, τ) and Ψ_t(x, τ). In order to distinguish them from one-argument functions they are denoted with subscript „ t ” Φ_t (x, τ) = Φ (x + τ) and Ψ_t (x, τ) = Ψ (x - τ). Selecting fixed time point of τ0 , it can be noted that: Φ_t (x, τ)= Φ ((x + τ0) + (τ - τ0)= Φ_t ((x + τ0, τ - τ0)) and Ψ_t (x, τ)= Ψ ((x - τ0) - (τ - τ0)= Ψ_t ((x - τ0, τ - τ0)). This means that values of function Φ() for position x and time τ are the same as in earlier time τ - τ0 for further position x + τ0; values of function Ψ() for position x and time τ are the same as in earlier time τ - τ0, but for closer position x - τ0 . Thus, it can be seen that function Ψ() represents a wave moving towards increasing values of x, and function Φ() represents a wave moving towards deceasing values of x. We can also note additional regularities: Φ_t (x, τ)= Φ_t (x + τ, 0) = Φ_t (0, τ + x), Ψ_t (x, τ)= Ψ_t (x - τ, 0) = Ψ_t (0, (- (x - τ)), Φ_t (x, 0) = Φ_t (0, x) , Ψ_t (x, 0) = Ψ_t (0, - x ). It is not justified to conduct more in-depth analysis (than presented here) for homogeneous wave equation (2.3) without additional conditions; as it is shown below, solutions represent just boundary conditions with shifted argument for equation (2.17). Formal notations:
Φ_t (0, 0) = Φ_t (x = - τ, τ) , Ψ_t (0, 0) = Ψ_t (x = τ, τ )	(2.15)
do not show that values of functions for τ = 0 depend on values of functions for τ > 0 respectively for position x = -τ for function Φ () and x = τ for function Ψ() (which is erroneously suggested in some textbooks). This proves only that values of functions in these positions are the same as in position (0,0) and apply to waves “already moving” and to waves which sources are located in point (0,0). To correctly present relations (2. 15) such that an effect is on the left side and a cause on the right side, we should write: Φ_t (x = - τ, τ) = Φ_t (0, 0) , Ψ_t (x = τ, τ ) = Ψ_t (0, 0) It is not possible then to justify any suggestions for an existence of any unknown, so called “advanced” solutions of Maxwell’s equations.