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which in conjunction with Eqs(49, 15, 45) leads to
0=∆
(T s+ µc ρc + v · g) vn − Π D nj vj − Tf D n
−µc j D c,n + c
E × H + E D × H 0
+E 0 × H D
· n
+ 〈ρvn − j D n 〉∆µ. (114)
Employing Eq(105), we have
R sf = −T ∆fn = fn ∆T + 〈ρvn − j D n 〉∆µ
+〈ρc vn − j D c,n〉∆µc − ∆(vj Π D jn)
+∆ (v · g vn)+c ∆
E × H + E D × H 0
+E 0 × H D
· n, (115)
which is more suitably written as
where
R sf = fn ∆T + 〈ρc vn − j D c,n〉∆µc
+
〈vn gj〉−Π D jn + 1
4 σsf∆Ej
∆vt,j
+〈ρvn − j D n 〉 ∆µ eff
+c n ×〈E D − vn
× B〉 · ∆H
c 0
+c 〈H D + vn
× D〉×n · ∆E
c 0
+[n × jel,sf] · (n × E 0 ) , (116)
µ eff
≡ µ +
v2 n gn
(ρvn − jD n ) − ΠDnn vn
(ρvn − jD
. (117)
n )
This R sf yields 9 boundary conditions and the value of the surface current
jel,sf.
4.3 Interfaces involving Conductors
The essential difference to the boundary conditions considered until now is
the fact that the electric field D is no longer an independent variable. As a
direct result, neither are the two boundary conditions,
18