Page 134 - Electromagnetics

3.2.2 Boundary conditions
Boundary conditions for the electrostatic field. The boundary conditions found
for the dynamic electric field remain valid in the electrostatic case. Thus
ˆ n 12 × (E 1 − E 2 ) = 0 (3.32)

and
ˆ n 12 · (D 1 − D 2 ) = ρ s . (3.33)
Here ˆ n 12 points into region 1 from region 2. Because the static curl and divergence
equations are independent, so are the boundary conditions (3.32)and (3.33).
For a linear and isotropic dielectric where D = E, equation (3.33)becomes
ˆ n 12 · ( 1 E 1 − 2 E 2 ) = ρ s . (3.34)
Alternatively, using D = 0 E + P we can write (3.33)as
1
ˆ n 12 · (E 1 − E 2 ) = (ρ s + ρ Ps1 + ρ Ps2 ) (3.35)
0
where

ρ Ps = ˆ n · P
is the polarization surface charge with ˆ n pointing outward from the material body.
We can also write the boundary conditions in terms of the electrostatic potential. With
E =−∇ , equation (3.32)becomes

1 (r) = 2 (r) (3.36)
for all points r on the surface. Actually 1 and 2 may differ by a constant; because
this constant is eliminated when the gradient is taken to find E, it is generally ignored.
We can write (3.35)as

∂ 1 ∂ 2
0 − =−ρ s − ρ Ps1 − ρ Ps2
∂n ∂n
where the normal derivative is taken in the ˆ n 12 direction. For a linear, isotropic dielectric
(3.33)becomes
∂ 1 ∂ 2
1 − 2 =−ρ s . (3.37)
∂n ∂n
Again, we note that (3.36)and (3.37)are independent.

Boundary conditions for steady electric current. The boundary condition on the
normal component of current found in § 2.8.2 remains valid in the steady current case.
Assume that the boundary exists between two linear, isotropic conducting regions having
constitutive parameters ( 1 ,σ 1 ) and ( 2 ,σ 2 ), respectively. By (2.198) we have
(3.38)
ˆ n 12 · (J 1 − J 2 ) =−∇ s · J s
where ˆ n 12 points into region 1 from region 2. A surface current will not appear on the
boundary between two regions having finite conductivity, although a surface charge may
accumulate there during the transient period when the currents are established [31]. If
charge is influenced to move from the surface, it will move into the adjacent regions,

© 2001 by CRC Press LLC