CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 128

Therefore, from the mathematical viewpoint, the problem consists of solv-
ing Eq. (11g) with the restriction Eq. (11f), which matches solving Eqs.
(11b) and (11d); in other words,

 

 





, , and (12)

∇× BEt = ∂ / ∂

∇× EBt = −∂ / ∂

∇⋅ B = 0

∇⋅ E = 0

1 2

in the MKS system; remembering that

( )

−

.

c = εμ = 1

00

The Faraday matrix representation becomes

0 BBE a −−

⎡ ⎤ : rows

z y x

⎢ ⎥

− B 0 B − E

⎢ ⎥

z x y

⎡

ab

⎤ =

(13a)

F

⎣ ⎦

⎢ ⎥

BB − 0 − E

y x z

⎢ ⎥

EEE 0

b : columns

⎢ ⎥

⎣ ⎦

x y z

And with Eq. (11c), an associated matrix for the dual tensor can be

constructed:

0 EEB −

⎡ ⎤

z y x

⎢ ⎥

− E 0 E B

⎢ ⎥

z x y

* ab

⎡ F ⎤ =

(13b)

⎣ ⎦

⎢ EE B

BBB

− 0 ⎥

y x z

⎢ ⎥

−−− 0

⎢ ⎥

⎣ ⎦

x y z

Note that Eq. (13b) is obtained if we make the following changes to Eq.
(13a):

  , EB  

(13c)

→−

BE →

Therefore, it may come to mind that * executes operation Eq. (13c); then,
it is clear that **

. Comparing Eqs. (11a) and (13a), we obtain the
relationship of the electric and magnetic fields with four-potential:

ar ar
FF =−

 

 

, B (14)

E =−∇ −∂ φ A / ∂ t

=∇× A

, where A 



() ( ) because

r AA

= , φ

and φ are the magnetic and electric poten-

tials, respectively.