Page 190 - Maxwell House

170 Chapter 4

energy. Simultaneously, through Maxwell’s equations (4.1) it is coupled to the magnetic field
and its energy too. Evidently, the same is right for the scalar potential. However, it does not
change the significant fact that both potentials are associated to EM energy concentrated into
electric and magnetic fields. As we will demonstrate later, both potentials describe the wave
propagating in space-time domain meaning they describe the transfer EM energy in space and
time.
More detailed discussion of the vector and scalar potentials and their physical significance is
beyond the classical electrodynamics and the scope of this book. The readers can find more
information attending [1 - 3]. In engineering practice, both potentials can be considered as a
very useful tool simplifying the analysis of electromagnetic waves generated by the sources.

4.1.5 Symmetry of Maxwell’s Equations and Principle of Duality. Electrodynamic
Potentials for Magnetic Sources

Although so far the magnetic charges and currents were nor observed, their introduction is very
useful engineering tool since the magnetic current magnitude is measured in Volts (see (1.10)).
As we have pointed out in Chapter 1, it will be perfectly valid without any mystics to interpret
some voltage sources of electromagnetic fields and field sensors as magnetic charges, dipoles,
and currents if the fields created by them and equivalent magnetic sources are identical. A tiny
loop with electric current as a magnetic dipole is exemplary.

Now suppose that we have gotten some solution of equation (4.22) with the given electric
current source. It is natural to ask how we can transform this solution if the source is magnetic
current. The elegant way to do so is the formal replacements of letters in the gotten solution
(principle of duality)

⟹ , ⟹ −, Σ ⟹ Σ , Σ ⟹ Σ, ⟺ (4.25)
0

As the following diagram demonstrates, Maxwell’s equations and Lorentz’s gauge (4.21)
(check please) are invariant under this transformation while electric source is replaced by
magnetic one

× = − × =

0
⎫ ⎧
⎪ ⎪
× = + ⎪ ⎪ × = − − Σ

0
Σ

⇒ (4.26)
∘ = ⁄ ⎬ ⎨ ∘ = Σ ⁄
0

Σ
∘ = 0 ⎪ ⎪ ∘ = 0
Σ ⎪ ⎪ Σ
∘ Σ + = 0 ⎭ ⎩ ∘ Σ + = 0

Therefore, we can acquire the new set of solutions of Maxwell’s equations
= −(1 ) × (4.27)
⁄

solving the wave equations of exact the same type as (4.22) and (4.23)
2
2 1
∇ − − = −

2 2 � (4.28)
2
2
1
∇ − − = −

2 2 0