ABSTRACT
Eq.(13.25), and reduces to the conservative field (11.10) in the static case. It is noteworthy that Eq.(13.23) is consistent with Eq.(12.23) since taking the divergence of each side of Eq.(13.23) gives
( ) ( )B dtdt
BE GGG ⋅∇∂−=⎟⎟⎠
⎞ ⎜⎜⎝ ⎛ ∂−⋅∇=×∇⋅∇
where both sides vanish, the left-hand side because divergence of curl is always zero and the right-hand side because of Eq.(12.23). 4. Ampère's law in a magnetic material modified by Maxwell as
dt DjH GGG ∂+=×∇ (14.1)
In the case where the fields vary with time, Ampère's circuital law (13.11) is inconsistent with the equation of continuity (12.15), since on applying the divergence to both sides, Eq.(13.11) becomes ( ) jH GG ⋅∇=×∇⋅∇ where the left-hand side vanishes identically. Maxwell assumed that the right hand side of Ampère's law should be written as a modified current density whose divergence must be zero. Its form was obtained by combining Eq.(11.26) with the equation of continuity (12.15), yielding
( ) 0or0 =⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
∂ ∂+⋅∇=⋅∇∂
∂+⋅∇ t DjD
t j
GGGG
This equation indicates that a current density /j D t+ ∂ ∂GG rather than jG should be taken on the right-hand side of Ampère's law, leading to Eq.(14.1). This corrected form of Ampère's law is consistent with the conservation of charge and reduces to the original circuital law (13.11) for a stationary charge flow, which obeys Eq.(12.16). The new term is called the displacement current density, the source of which is a time-varying electric displacement .D
G
EXAMPLE 14.1. The displacement current in a capacitor The existence of a displacement current is suggested by the case of an alternating current applied to the plates of a capacitor, as illustrated in Figure 14.1. There is no charge transfer between the plates, however a current passes through the capacitor. Drawing two surfaces S and S ′ bound to the same closed path, Ampère's circuital law (12.26) involving the current through S reads
S'
S
Γ
Figure 14.1. A capacitor in an alternating current circuit For the uniform field Erε between the plates, Gauss's theorem gives
This displacement current through 'j
G S ′ originates in the time rate of change of the
displacement vector, which is associated with the change in the surface charge densities on the plates. We have seen that taking the divergence of Eqs.(13.23) and (14.1) yields equations that are consistent with Eqs.(12.23) and (11.26), respectively, so that the four Maxwell's field equations form two pairs of differential equations, which can be written in terms of free charges and free currents in materials as
dt DjH
dt BE
GGGGG ∂+=×∇∂−=×∇ (14.2)
0B D ρ∇ ⋅ = ∇ ⋅ =G G or in integral form
E dl B dS tΓ
∂⋅ = − ⋅∂∫ ∫ G GG G
DH dl j dS tΓ
⎛ ⎞∂⋅ = + ⋅⎜ ⎟∂⎝ ⎠∫ ∫ GG GG G
(14.3) 0
B dS⋅ =∫ GG V
V S
D dS dρ⋅ =∫ ∫GG Both forms of Eqs.(14.2) and (14.3) are valid for arbitrary media, since they contain no parameters other than the free-charge density and free-current density.
