Page 64 - Electromagnetics Handbook

and for the moving frame
∂B

E · dl =− · dS , (2.143)
S ∂t

∂D

H · dl = · dS + J · dS . (2.144)
S ∂t S

Here boundary contour has sense determined by the right-hand rule. We use the

notation , S , etc., to indicate that all integrations for the moving frame are computed
using space and time variables in that frame. Equation (2.141) is the integral form of
Faraday’s law, while (2.142) is the integral form of Ampere’s law.
Faraday’s law states that the net circulation of E about a contour (sometimes called
the electromotive force or emf ) is determined by the flux of the time-rate of change of the
flux vector B passing through the surface bounded by . Ampere’s law states that the
circulation of H (sometimes called the magnetomotive force or mmf ) is determined by
the flux of the current J plus the flux of the time-rate of change of the flux vector D.It is
the term containing ∂D/∂t that Maxwell recognized as necessary to make his equations
consistent; since it has units of current, it is often referred to as the displacement current
term.
Equations (2.141)–(2.142) are the large-scale or integral forms of Maxwell’s equations.
They are the integral-form equivalents of the point forms, and are form invariant under
Lorentz transformation. If we express the fields in terms of the moving reference frame,
we can write
d

E · dl =− B · dS , (2.145)
dt
S
d

H · dl = D · dS + J · dS . (2.146)
dt S S
These hold for a stationary surface, since the surface would be stationary to an observer
who moves with it. We are therefore justified in removing the partial derivative from the
integral. Although the surfaces and contours considered here are purely mathematical,
they often coincide with actual physical boundaries. The surface may surround a moving
material medium, for instance, or the contour may conform to a wire moving in an
electrical machine.
We can also convert the auxiliary equations to large-scale form. Consider a volume
region V surrounded by a surface S that moves with velocity v relative to the laboratory
frame (Figure 2.3). Integrating the point form of Gauss’s law over V we have

∇· D dV = ρ dV.
V V
Using the divergence theorem and recognizing that the integral of charge density is total
charge, we obtain

D · dS = ρ dV = Q(t) (2.147)
S V
where Q(t) is the total charge contained within V at time t. This large-scale form of
Gauss’s law states that the total flux of D passing through a closed surface is identical
to the electric charge Q contained within. Similarly,

B · dS = 0 (2.148)
S

© 2001 by CRC Press LLC