Page 96 - Electromagnetics Handbook

they are the mechanism for exerting force on the charge distribution. Substituting for J
from (2.2) and for ρ from (2.3) we have

∂D
f em = E(∇· D) − B × (∇× H) + B × .
∂t
Using

∂D ∂ ∂B
B × =− (D × B) + D ×
∂t ∂t ∂t
and substituting from Faraday’s law for ∂B/∂t we have
∂
− [E(∇· D) − D × (∇× E) + H(∇· B) − B × (∇× H)] + (D × B) =−f em . (2.287)
∂t
Here we have also added the null term H(∇· B).
The forms of (2.287) and (2.279) would be identical if the bracketed term could be
¯
written as the divergence of a dyadic function T em . This is indeed possible for linear,
¯
homogeneous, bianisotropic media, provided that the constitutive matrix [C EH ] in (2.21)
is symmetric [101]. In that case
1
¯ ¯
T em = (D · E + B · H)I − DE − BH, (2.288)
2
which is called the Maxwell stress tensor. Let us demonstrate this equivalence for a
linear, isotropic, homogeneous material. Putting D =
E and H = B/µ into (2.287) we
obtain
1 1
∇· T em =−
E(∇· E) + B × (∇× B) +
E × (∇× E) − B(∇· B). (2.289)
µ µ
Now (B.46) gives
∇(A · A) = 2A × (∇× A) + 2(A ·∇)A

so that
1 2
E(∇· E) − E × (∇× E) = E(∇· E) + (E ·∇)E − ∇(E ).
2
Finally, (B.55) and (B.63) give
1

¯
E(∇· E) − E × (∇× E) =∇ · EE − IE · E .
2
Substituting this expression and a similar one for B into (2.289) we have
1

¯ ¯
∇· T em =∇ · (D · E + B · H) I − DE − BH ,
2
which matches (2.288).
¯
Replacing the term in brackets in (2.287) by ∇· T em , we get
¯ ∂g em
∇· T em + =−f em (2.290)
∂t
where
g em = D × B.

© 2001 by CRC Press LLC