Page 77 - Electromagnetics Handbook

The tangential magnetic field across a thin source distribution is discontinuous by an
amount equal to the surface current density.
Similar steps with Faraday’s law give
ˆ n 12 × (E 1 − E 2 ) = 0.

The tangential electric field is continuous across a thin source.
We can also derive conditions on the normal components of the fields, although these
are not required for uniqueness. Gauss’s law (2.147) applied to the volume V in Figure
2.5 gives

D 1 · ˆ n 1 dS + D 2 · ˆ n 2 dS + D · ˆ n 3 dS = ρ dV.
V
S 1 S 2 S 3
As → 0, the thin source layer recedes to a surface layer. The integral of normal D over
S 3 tends to zero by continuity of the fields. By symmetry S 1 = S 2 and ˆ n 1 =−ˆ n 2 = ˆ n 12 .
Thus

(D 1 − D 2 ) · ˆ n 12 dS = ρ dV. (2.183)
S 1 V
The volume integral is
δ/2 δ/2
ρ dV = ρ dS dx = f (x, ) dx ρ s (r, t) dS.
V S 1 −δ/2 −δ/2 S 1
Since δ = k has been chosen so that most of the source charge lies within V , (2.181)
gives

[(D 1 − D 2 ) · ˆ n 12 − ρ s ] dS = 0,
S 1
hence
(D 1 − D 2 ) · ˆ n 12 = ρ s .

The normal component of D is discontinuous across a thin source distribution by an
amount equal to the surface charge density. Similar steps with the magnetic Gauss’s law
yield
(B 1 − B 2 ) · ˆ n 12 = 0.
The normal component of B is continuous across a thin source layer.
We can follow similar steps when a thin magnetic source layer is present. When
evaluating Faraday’s law we must include magnetic surface current and when evaluating
the magnetic Gauss’s law we must include magnetic charge. However, since such sources
are not physical we postpone their consideration until the next section, where appropriate
boundary conditions are postulated rather than derived.
2.8.2 Boundaryconditions across a stationarylayer of field disconti-
nuity
Provided that we model a surface source as a limiting case of a very thin but continuous
volume source, we can derive boundary conditions across a surface layer. We might ask
whether we can extend this idea to surfaces of materials where the constitutive parameters
change from one region to another. Indeed, if we take Lorentz’ viewpoint and visualize a
material as a conglomerate of atomic charge, we should be able to apply this same idea.
After all, a material should demonstrate a continuous transition (in the macroscopic

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