10.2 The Euler Equations of Functionals …
629
If J s = 0, then Eq. (4) can be written as
∇ × H = σ E + ε
∂ E
∂t
(17)
Taking the rotation to Eq. (5), and using B = μH, we obtain
∇ × ∇ × E = −∇ × μ
∂ H
∂t
= −μ
∂
∂t
(∇ × H)
(18)
Using the rotation operation formula ∇ ×∇ × E = ∇(∇ · E)−E and Eqs. (17),
(18) can be transformed to
∇(∇ · E) − E = −μσ
∂ E
∂t
− με
∂
2 E
∂t 2
(19)
Substituting D = ε E into Eq. (7), then substituting Eq. (7) into Eq. (19), we get
E = μσ
∂ E
∂t
+ με
∂
2 E
∂t 2 − ∇
ρ
ε
(20)
When the free charge density ρ = 0, there is
E = μσ
∂ E
∂t
+ με
∂
2 E
∂t 2
(21)
Taking the rotation to Eq. (17), using the rotation operation formula ∇ ×∇ × H =
∇(∇ · H) − H again, and using B = μH and Eq. (5), we get
H = μσ
∂ H
∂t
+ με
∂
2 H
∂t 2
(22)
Equations (21) and (22) have the same form, the two equations are called the
telegraph equation.
Making the inner product of Eq. (4) and E, the inner product of Eq. (5) and H,
we get
E · ∇ × H = E · J + E ·
∂ D
∂t
(23)
H · ∇ × E = −H ·
∂ B
∂t
(24)
where, E · J is the energy density of heat consumption, E ·
∂ D
∂t
is the rate of change
that the electric field energy density varies with time, H ·
∂ B
∂t
is the rate of change
that the magnetic field energy density varies with time.