An Assay
Mathematics Background of the Maxwell equations
Let us consider Maxwell equations system:
(1)
(2)
This is a system of the two dependent variables, x and y, which generally depend on independent scalar, i. e. t (say, it be ‘time’).
In the matrix form it is:
dX/dt = A*X, (3)
where X = (x; y) is a column of the two coordinates, x and y, on the Descartes’ plane;
A is a 2 x 2 matrix of scalars: a11, a12, a21, a22,
Where a11 = 0; a12 = –1; a21 = 1; a22 = 0.
There we understand A*X as a product of matrices A (a 2 x 2 matrix) and X (a 1 x 2 matrix, or a column two-component vector of components x and y).
In addition, t is an independent variable (time).
The solution of this system shall be:
X = exp(A*t). (4)
So we can see that Maxwell equations (in the three-dimensional space can be simplified to the following system of 2 equations, in the two-dimensional space:
“Maxwell's 4 equations simplify down to 2 equations (considering a single spatial dimension):dE/dx = -dB/dt dE/dt = -c2 dB/dx (5)So when the temporal derivative of one is maximal the spatial derivative of the other is minimal (maximally negative). If you consider a simple single-frequency sinusoidal plane wave, you find that this happens for E and B in phase. In the above equations:E = Emax cos(kx-wt)B = Bmax cos(kx-wt)” [1] (6)
I we consider there, in addition, the phase, the first equation from the above cited should be:
E = Emax cos(kx-wt+φ), (7)
Or, which is the same,
E = Emax cos(kx-wt+phase), (8)
Where phase = φ that means the phase of oscillations taken along the x axis, i. e. ‘displacement’ of a sinusoid along the x axis.
E is the current value of the electric field intensity;
Emax is the maximum value of the electric field intensity (the oscillations’ amplitude);
cos means the cosine math function;
k is a quotient (depending of physical constants, e.g. the light speed);
x is a coordinate on the line (here it is a ‘1-dimensional space, a line);
w is the radial frequency of the oscillations;
t is time.
Maxwell discovered the equations in 1861.
Since this time, they have been actively discussed and explored.
Review of literature
For instance we can cite the following excerpt:
“Analysis of Maxwell's Equations
Maxwell's equations form the backbone of modern electrodynamics theories. The 20 equations originally presented by Clerk Maxwell were altered and reduced to four equations by Oliver Heaviside. This may sound conspiratorial to some alternative science buffs, but there is a good reason for the rewrite. Maxwell's math and reasoning were seriously flawed in several instances.
However, Oliver Heaviside didn't catch the most crucial flaw, which is the dimension of charge is incorrectly notated in all systems of units except the centimeter-gram-second (cgs) system. The cgs system of units was not developed with knowledge of the dimension of charge, but it was implied because charge as a dimension is a fact of nature, not an invention of science.” [2]
Maxwell Equations and Everyday Life
We meet with the processes, described by the Maxwell equations, in our everyday life. The equations most generally describe wave transmission in the surrounding space. Merely, the Maxwell equations rely to the electromagnetic oscillations (waves). The light, Roentgen rays, radio range waves, heat irradiation are all electromagnetic oscillations (waves). All of use radio sets, satellite communications, GPRS, microwave ovens, Roentgen medical investigations etc. All of them based on electromagnetic waves first considerably explained by the great Scottish physicist Maxwell in 1861.
References:
[1] https://geektimes.ru/post/271494/
[2] http://news.james-clerk-maxwell.com/en/
[3] http://www.geo.mtu.edu/~scarn/teaching/GE4250/ComplexWave_lecture.ppt
