Page 74 - Electromagnetics Handbook

3
C and current in C/s. The physical symmetry is equally appealing: magnetic flux lines
diverge from magnetic charge, and the total flux passing through a surface is given by the
total magnetic charge contained within the surface. This is best seen by considering the
large-scale forms of Maxwell’s equations for stationary surfaces. We need only modify
(2.145) to include the magnetic current term; this gives

d
E · dl =− J m · dS − B · dS, (2.173)
S dt S
d

H · dl = J · dS + D · dS. (2.174)
S dt S
If we modify (2.148) to include magnetic charge, we get the auxiliary equations

D · dS = ρ dV,
S V

B · dS = ρ m dV.
S V
Any of the large-scale forms of Maxwell’s equations can be similarly modified to include
magnetic current and charge. For arbitrarily moving surfaces we have
d

∗ ∗
E · dl =− B · dS − J · dS,
m
(t) dt S(t) S(t)

d
∗ ∗
H · dl = D · dS + J · dS,
(t) dt S(t) S(t)
where
∗
E = E + v × B,
H = H − v × D,
∗
J = J − ρv,
∗
J = J m − ρ m v,
∗
m
and all fields are taken to be measured in the laboratory frame with v the instantaneous
velocity of points on the surface and contour relative to the laboratory frame. We also
have the alternative forms

∂B
(ˆ n × E) dS = − − J m dV, (2.175)
∂t
S V

∂D

(ˆ n × H) dS = + J dV, (2.176)
S V ∂t
and

d
[ˆ n × E − (v · ˆ n)B] dS =− J m dV − B dV, (2.177)
S(t) V (t) dt V (t)
d

[ˆ n × H + (v · ˆ n)D] dS = J dV + D dV, (2.178)
dt
S(t) V (t) V (t)
3 We note that if the modern unit of T is used to describe B, then ρ m is described using the more
cumbersome units of T/m, while J m is given in terms of T/s. Thus, magnetic charge is measured in Tm 2
2
and magnetic current in (Tm )/s.
© 2001 by CRC Press LLC