_{Quotations from Evans' book [1] in} black

In Chap. 2.3 of his book [1] Evans gives an equation for the B⁽³⁾ field that according to his B⁽³⁾ hypothesis is associated to a circularly polarized plane wave:

                B(3) = eμoc/h- I/ω² e(3) = 5.723 × 1027 I/ω² e(3)                         (137)

where [1, p.1] e is the charge of an electron, m its mass, h^- is the Dirac constant and [1, p.3] e⁽³⁾=k is a unit vector in the (3) axis of wave propagation. The plane wave B is derived from its vector potential A [1, p.34]:

In S.I. units the fundamental equation linking A to the magnetic field B is in classical electrodynamics [47],

                B = Ñ×A                                                                                 (127)

So if A is a plane wave in vacuo then so is B (and its electric counterpart E). If the plane wave A is a solution of the vacuum d'Alembert equation then it may be written as

                A⁽¹⁾ = A^(2)* = ^A(o)/_2^½ (ii+j) e^iΦ                                                 (128)

From Eq. (127), the plane wave is

                B⁽¹⁾ = B^(2)* = ^ω/_c A⁽¹⁾ = ^B(o)/_2^½ (ii+j) e^iΦ                                 (129)

. . .

Here Φ is the electromagnetic phase [1,2]. A^(o), B^(o) ... are scalar amplitudes, and i and j are unit Cartesian vectors in X and Y, perpendicular to the propagation direction Z of the wave. The following key relations then follow using elementary algebra,

                A⁽¹⁾×A⁽²⁾ = ^c²/_ω² B⁽¹⁾×B⁽²⁾ = ¹/_ω² E⁽¹⁾×E⁽²⁾                                 (131)

and show that the product A⁽¹⁾×A⁽²⁾ is proportional to B⁽¹⁾×B⁽²⁾ divided by the square of the angular frequency. Expressing B⁽¹⁾×B⁽²⁾ in terms of the beam intensity or the power density (I in W/m²)

                B⁽¹⁾×B⁽²⁾ = i ^μ_o/_c I e^(3)*                                                                 (132)

where μ_o is the vacuum permeability in S.I. (Chap. 1).

The most important part of Evans' B⁽³⁾ hypothesis is the alleged symmetry relation [1, p.121]

                B⁽¹⁾×B⁽²⁾ = i B^(o) B^(3)* = i B^(o)² e^(3)* ,             et cyclicum,               (357)

(see also [1, eqs.(86), (154) and (181)])

which, of course, if Evans should be right, must be fulfilled in addition.

From this hypothesis together with Evans' eq. [1,(132)] we obtain the relation

                I = ^c/_μo B^(o)² ,                                                                                 (B)

(c.f. [1,eq.(183)]) which shows that at constant power density I the values of |B⁽³⁾| = B^(o), i.e. the entries of the second column of TABLE 2 should be constant as well. Evans assumes I = 1 kW/m² for the TABLE 2 in [1, p.37]), which due to eq. (B) yields

                B^(o) = 65 microTesla                 for arbitrary frequency ω .

However, this is a contradiction to the radiation formula [1, eq.(137)] which yields the non-constant values in the second column of TABLE 2 for 7 sample values of ω in the first column.

Evans' B(3) symmetry hypothesis [1, eq.(357)]) and his radiation formula [1, eq.(137)] disagree.