Direct Energy, 2018a

280 12.3 Mechanical Energy Conversion
Interestingly, there is a close relationship between the quantities in Tables
12.3 and 12.5. Maxwell's equations, rst introduced in Section 1.6.1,
relate the four electromagnetic eld parameters. Assuming no sources,
−→ J =0and ρch =0, Maxwell's equations can be written:
−→ ∇×
−→ E = −
∂ −→ B
∂t
−→ ∇×
−→ H =
∂ −→ D
∂t
(12.10)
(12.11)
−→ ∇·−→ D =0 (12.12)
−→ ∇·−→ B =0 (12.13)
The last two relationships, Gauss's laws, result directly from using calculus
of variations to set up the Euler-Lagrange equation and solving for the
corresponding equation of motion. We can replace electromagnetic vector
elds in the source-free version of Maxwell's equations by mechanical elds
according to the transformation:
−→ D →
−→ M (12.14)
−→ E →
−→ v (12.15)
−→ B →
−→ τ (12.16)
−→ H →
−→ θ (12.17)
The transformation of Eqs. 12.14 -12.17 leads to set of equations accurately
describing relationships between these mechanical elds.
−→ ∇×
−→ v = −
∂ −→ θ
∂t
−→ ∇×
−→ τ =
∂ −→ M
∂t
(12.18)
(12.19)
−→ ∇·−→ M =0 (12.20)
−→ ∇·−→ θ =0 (12.21)