September 04, 2008, updated on September 12, 2008

_{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 × 10^{27} ^{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)}/_{2½}
(*i***i**+**j**) e^{iΦ}
(128)

From Eq. (127), the plane wave is

**B**^{(1)} = **B**^{(2)*} =
^{ω}/_{c} **A**^{(1)} =
^{B(o)}/_{2½}
(*i***i**+**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* ^{μo}/_{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.

[1] M.W. Evans e.a., *The Enigmatic Photon*, Vol.3,
KLUWER 1996