Page 82 - Electromagnetics Handbook

2.8.4 Boundaryconditions across a stationarylayer of field disconti-
nuityusing equivalent sources

So far we have avoided using the physical interpretation of the equivalent sources in the
Maxwell–Boffi equations so that we might investigate the behavior of fields across true
discontinuities. Now that we have the appropriate boundary conditions, it is interesting
to interpret them in terms of the equivalent sources.
If we put H = B/µ 0 − M into (2.194) and rearrange, we get
ˆ n 12 × (B 1 − B 2 ) = µ 0 (J s + ˆ n 12 × M 1 − ˆ n 12 × M 2 ). (2.210)

The terms on the right involving ˆ n 12 ×M have the units of surface current and are called
equivalent magnetization surface currents. Defining

J Ms =−ˆ n × M (2.211)

where ˆ n is directed normally outward from the material region of interest, we can rewrite
(2.210) as

ˆ n 12 × (B 1 − B 2 ) = µ 0 (J s + J Ms1 + J Ms2 ). (2.212)
We note that J Ms replaces atomic charge moving along the surface of a material with an
equivalent surface current in free space.
If we substitute D =
0 E + P into (2.196) and rearrange, we get

1
ˆ n 12 · (E 1 − E 2 ) = (ρ s − ˆ n 12 · P 1 + ˆ n 12 · P 2 ). (2.213)

0
The terms on the right involving ˆ n 12 · P have the units of surface charge and are called
equivalent polarization surface charges. Defining

ρ Ps = ˆ n · P, (2.214)
we can rewrite (2.213) as

1
ˆ n 12 · (E 1 − E 2 ) = (ρ s + ρ Ps1 + ρ Ps2 ). (2.215)

0
We note that ρ Ps replaces atomic charge adjacent to a surface of a material with an
equivalent surface charge in free space.
In summary, the boundary conditions at a stationary surface of discontinuity written
in terms of equivalent sources are

ˆ n 12 × (B 1 − B 2 ) = µ 0 (J s + J Ms1 + J Ms2 ), (2.216)
ˆ n 12 × (E 1 − E 2 ) =−J ms , (2.217)
1
ˆ n 12 · (E 1 − E 2 ) = (ρ s + ρ Ps1 + ρ Ps2 ), (2.218)

0
ˆ n 12 · (B 1 − B 2 ) = ρ ms . (2.219)

2.8.5 Boundaryconditions across a moving layer of field discontinuity
With a moving material body it is often necessary to apply boundary conditions de-
scribing the behavior of the fields across the surface of the body. If a surface of discon-
tinuity moves with constant velocity v, the boundary conditions (2.194)–(2.199) hold as

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