For a Vacuum Solution, as shown in Geometric Algebra for Electrical
and Electronic Engineers by IEEE, and other places, Electric Field is shown as a Helix. And M (though being Axial Vector), in terms of logical wave, it is again a Helxi by phase difference of 90 degrees.
If the propagation is z-direction, the x-z has Sin projection and y-z has Cos Projection for E. And Magnetic it is Cos and -Sin. We will take Em and Bm =1 for magnitudes. We take c = 1, e = 1, u = 1. So B = H.
One can model Y-Z Rotations has function of angular frequency and Z as displacement from momentum.
Ex = Sin (wt), Ey = Cos (wt), and Ez = t
Bx = Cos(wt), By = -Sin (wt), and Bz = t.
S = H x B gives
Sx = Hy x Ez - Hz x Ey = -t[Sin(wt) + Cos(wt)] = -t 2 ^1/2(Sin(wt + 45))
Sy = Hz x Ex - Hx x Ez = t[Sin(wt) - Cos (wt)] = t 2 ^1/2(Sin(wt - 45))
Sz = Hx x Ey - Hy x Ez = Cos^2 (wt) + Sin ^2 (wt) = 1
So with increasing "t", the Pyonting Vector will spread in X and Y direction but not remain constant in Z direction!
What is wrong with my understanding?
If you only consider radiation from one point, then it does indeed spread out as you have found. If you consider the combined radiation from all the points across a plane wave, then you will find that the contributions add and subtract in the right places to give a plane wave carrying on. At the edges, where some of the contributions needed to make the plane wave are missing, the plane wave breaks down. This is called diffraction.
Read about Huygens' description of wave propagation. The Fresnel and Fraunhoffer descriptions of wave propagation are also relevant.
Can you please help me with a related math? I am having a problem with justifying the above solution (with some minor correction) to satisfy Maxwell Equations for vacuum.
Solution to Maxwell Equations and Poynting Vector Point to Dispersion Geometry.
For a Vacuum Solution for EMF propogation, as shown in Geometric Algebra for Electrical and Electronic Engineers by IEEE, and other places, the Electric Field is shown as a Helix. And the Magnetic Field, though being Axial Vector, is again a Helix with a phase difference of 90 degrees.
If the propagation is z-direction, then one may assume the x-z plane has the Sin projection and the y-z plane has the Cos Projection for the Electric Field. And the Magnetic Field has the Cos and the -Sin projections. We will take their amplitudes to be 1, along c = 1, Є = 1, µ = 1. So, B = M but in reality, they will have different spans.
One can model the x-y plane has rotations with an angular frequency of “ɯ” and the Z as displacement from the momentum.
Ex = Sin (ɯt), Ey = Cos (ɯt), and Ez = t
Bx = Cos(ɯt), By = -Sin (ɯt), and Bz = t.
S = E x B gives
Sx = Ey Bz - Ez By = t[Sin(ɯt) + Cos(ɯt)] = √2 t (Sin(ɯt + 45))
Sy = Ez Bx - Ex Bz = t[- Sin(ɯt) + Cos (ɯt)] = - √2 t (Cos(ɯt + 45))
Sz = Ex By - Ey Bx = - Cos^2 (ɯt) - Sin ^2 (ɯt) = -1
So with increasing "t", the Pyonting Vector will spread in X and Y direction but not remain constant in Z direction
I am unable to prove Del Operator x E = - dB/dt for Vaccum. :-( 60 years old with many health issues. :-(
Malcolm White
I suggest you read the Wikipedia article https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle and the assciated links. That might help.
Only radiation from an infinitely wide wave front doesn't spread out and attenuate.
My question is about solution detail in satisfying Maxwell Solution. I am getting one extra term in time derivative in B and not in curl of E - and they need to match.
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