10.2 The Euler Equations of Functionals …
627
Using of the identities of vector operation ∇ ×∇ϕ = 0 and ∇ ·∇ × A = 0, taking
the rotation to E = −∇ϕ −
∂ A
∂t
, and taking the divergence to B = ∇ × A, we obtain
−∇ × ∇ϕ = ∇ × E +
∂(∇× A)
∂t
= ∇ × E +
∂ B
∂t
= 0, ∇ · B = ∇ · (∇ × A) = 0, that is
∇ × E = −
∂ B
∂t
(5)
∇ · B = 0
( 6 )
Equation (3) can be written as
∇ · D = ρ
(7)
The four equations from Eqs. (4) to (7) are called the Maxwell(’s) equations in
differential form in the electromagnetic field with media or Maxwell(’s) equations
for a time-varying electromagnetic field. Customarily, the four equations are called
the first, second, third, fourth Maxwell(’s) equation in turn. The first Maxwell’s
equation is also called the Maxwell-Ampère circulation law. The second Maxwell’s
equation is also called the Faraday law of electromagnetic induction. The third
Maxwell’s equation is also called the Gauss theorem of magic field. The fourth
Maxwell’s equation is also called the Gauss theorem of electric field. Of course,
the Maxwell’s equations have also other forms. In vacua, μ r = 1, ε r = 1, if μ and ε
in the Maxwell’s equations are changed into μ 0 and ε 0 , then Eqs. (4)–(7) are called
the Maxwell(’s) equations for a time-varying electromagnetic field in vacua.
Taking the divergence to Eq. (4) and making use of Eq. (7), we have
∇ · (∇ × H) = ∇ · J + ∇ ·
∂ D
∂t
= ∇ · J +
∂∇ · D
∂t
= ∇ · J +
∂ρ
∂t
= 0
(8)
or
∇ · J = −
∂ρ
∂t
(9)
Equation (9) is called the conservation law of charge in differential form or the
continuity equation of electric current. This shows that the continuity equation of
electric current can be derived from the Maxwell equations.
In Eqs. (4)–(9), the vectors B, E, J and the scalar ρ are all functions of space and
time, and there exist relations D = ε E, H = B/μ, namely the vectors B and H
are also functions of space and time. Change B, D, E, H, J and ρ into the following
complex vectors and complex scalar
B(q 1 , q 2 , q 3 )e
iωt
, D(q 1 , q 2 , q 3 )e
iωt
, E(q 1 , q 2 , q 3 )e
iωt
H(q 1 , q 2 , q 3 )e
iωt
, J(q 1 , q 2 , q 3 )e
iωt
, ρ(q 1 , q 2 , q 3 )e
iωt