Page 81 - Electromagnetics Handbook

which we take as our postulate of the boundary condition on current across a surface
containing discontinuities. A similar set of steps carried out using the continuity equation
for magnetic sources yields
∂ρ ms
∇ s · J ms =−ˆ n 12 · (J m1 − J m2 ) − .
∂t
In summary, we have the following boundary conditions for fields across a surface
containing discontinuities:
ˆ n 12 × (H 1 − H 2 ) = J s , (2.194)
ˆ n 12 × (E 1 − E 2 ) =−J ms , (2.195)
ˆ n 12 · (D 1 − D 2 ) = ρ s , (2.196)
ˆ n 12 · (B 1 − B 2 ) = ρ ms , (2.197)
and
∂ρ s
ˆ n 12 · (J 1 − J 2 ) =−∇ s · J s − , (2.198)
∂t
∂ρ ms
ˆ n 12 · (J m1 − J m2 ) =−∇ s · J ms − , (2.199)
∂t
where ˆ n 12 points into region 1 from region 2.

2.8.3 Boundaryconditions at the surface of a perfect conductor
We can easily specialize the results of the previous section to the case of perfect electric
or magnetic conductors. In § 2.2.2 we saw that the constitutive relations for perfect
conductors requires the null field within the material. In addition, a PEC requires zero
tangential electric field, while a PMC requires zero tangential magnetic field. Using
(2.194)–(2.199), we find that the boundary conditions for a perfect electric conductor
are

ˆ n × H = J s , (2.200)
ˆ n × E = 0, (2.201)
ˆ n · D = ρ s , (2.202)
ˆ n · B = 0, (2.203)
and

∂ρ s
ˆ n · J =−∇ s · J s − , ˆ n · J m = 0. (2.204)
∂t
For a PMC the conditions are
ˆ n × H = 0, (2.205)
ˆ n × E =−J ms , (2.206)
ˆ n · D = 0, (2.207)
ˆ n · B = ρ ms , (2.208)
and
∂ρ ms
ˆ n · J m =−∇ s · J ms − , ˆ n · J = 0. (2.209)
∂t
We note that the normal vector ˆ n points out of the conductor and into the adjacent
region of nonzero fields.

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