§ 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?

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, 20178 yr by Vmedvil