Maxwell's Structure of Light

[reerer]

§ 1. Maxwell's Structure of Light

 

 

The electromagnetic transverse wave equations of light are derived using Maxwell's equations,

 

 

∇ x E = - dB/dt........................∇ x B = 1/c (dE/dt).....................................73a,b

 

 

Maxwell's curl equations (equ 73a,b) are expanded to form,

 

 

dE_z/dy - dE_y/dz = - dB_x/dt...........................................................................74

 

dE_x/dz - dE_z/dx = - dB_y/dt...........................................................................75

 

dE_y/dx - dE_x/dy = - dB_z/dt...........................................................................76

 

 ...........................................................

 

dB_z/dy - dB_y/dz = 1/c (dE_x/dt)....................................................................77

 

dB_x/dz - dB_z/dx = 1/c (dE_y/dt)....................................................................78

 

dB_y/dx - dB_x/dy = 1/c (dE_z/dt)..................................................... ..............79

 

The z-direction electric transverse wave equations is derived using equations 74 and 78 by eliminating dE_y/dz and dB_z/dx  to form (Jenkins, p. 410),

 

 

 

dE_y/dz = 1/c (dB_x/dt)..............................dB_x/dz = 1/c (dE_y/dt)...................80a,b

 

 

Differentiating equation 80a, with the respect to d/dz, and equation 80b with respect to d/dt produces (Condon, p, 1-108),

 

 

d²E_y/d²z = 1/c (d²B_x/dtdz)......................d²B_x/dtdz = 1/c (d²E_y/d²t)...........81a,b

 

 

Equating equations 81a,b,

 

 

d²E_y/d²z = 1/c² (d²E_y/d²t)...........................................................................82

 

 

Differentiating equation 82a, with the respect to d/dt, and equation 82b with respect to d/dz produces ,

 

 

d²E_y/dtdz = 1/c (d²B_x/d²t)......................d²B_x/d²z = 1/c (d²E_y/dtdz)...........83a,b

 

 

Equating equations 83a,b forms,

 

 

d²B_x/d²z = 1/c² (d²B_x/d²t)..........................................................................84

 

 

Equations 82 and 84 are used to derive the z direction electromagnetic transverse wave equations of light (fig 17),

 

 

E_y = E_o cos(kz - wt) ĵ ..............................................................................85

 

B_x = B_o cos(kz -wt) î ................................................................................86

 

 

In the derivation of equations 80a,b, 14 of the 18 differential components that constitute Maxwell's equations are eliminated since an electromagnetic field within a volume forms a horizontal wave.

 

 

What do you think of the mathematic that is being depicted?

[Vmedvil]

Um, Elimination doesn't work on differential equations, they are arrays unless you mean elimination in the array or matrix form to solve them.

Wait, I get what he is saying, but he gave it the wrong term.

Edited November 15, 2017 by Vmedvil