Page 80 - Electromagnetics Handbook

as the boundary condition appropriate to a surface of field discontinuity containing a
magnetic surface current.
We can also postulate boundary conditions on the normal fields to make Gauss’s laws
valid for surfaces of discontinuous fields. Integrating (2.147) over the regions V 1 and V 2
and adding, we obtain

D · ˆ n dS − D 1 · ˆ n 10 dS − D 2 · ˆ n 20 dS = ρ dV.
S 1 +S 2 S 10 S 20 V 1 +V 2
As δ → 0 this becomes

D · ˆ n dS − ρ dV = (D 1 − D 2 ) · ˆ n 12 dS. (2.191)
S V S 10
If we integrate Gauss’s law over the entire region V , including the surface of discontinuity,
we get

D · ˆ n dS = ρ dV + ρ s dS. (2.192)
S V S 10
In order to get identical answers from (2.191) and (2.192), we must have

(D 1 − D 2 ) · ˆ n 12 = ρ s
as the boundary condition appropriate to a surface of field discontinuity containing an
electric surface charge. Similarly, we must postulate

(B 1 − B 2 ) · ˆ n 12 = ρ ms
as the condition appropriate to a surface of field discontinuity containing a magnetic
surface charge.
We can determine an appropriate boundary condition on current by using the large-
scale form of the continuity equation. Applying (2.10) over each of the volume regions
of Figure 2.6 and adding the results, we have
∂ρ

J · ˆ n dS − J 1 · ˆ n 10 dS − J 2 · ˆ n 20 dS =− dV.
∂t
S 1 +S 2 S 10 S 20 V 1 +V 2
As δ → 0 we have
∂ρ

J · ˆ n dS − (J 1 − J 2 ) · ˆ n 12 dS =− dV. (2.193)
S S 10 V ∂t
Applying the continuity equation over the entire region V and allowing it to intersect
the discontinuity surface, we get
∂ρ ∂ρ s

J · ˆ n dS + J s · ˆ m dl =− dV − dS.
S V ∂t S 10 ∂t
By the two-dimensional divergence theorem (B.20) we can write this as

∂ρ ∂ρ s
J · ˆ n dS + ∇ s · J s dS =− dV − dS.
S S 10 V ∂t S 10 ∂t
In order for this expression to produce the same values of the integrals over S and V as
in (2.193) we require
∂ρ s
∇ s · J s =−ˆ n 12 · (J 1 − J 2 ) − ,
∂t

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