Page 67 - Electromagnetics Handbook

(The asterisk should not be confused with the notation for complex conjugate.)
Much confusion arises from the similarity between (2.153) and (2.145). In fact, these

expressions are different and give different results. This is because B in (2.145) is
measured in the frame of the moving circuit, while B in (2.153) is measured in the frame
of the laboratory. Further confusion arises from various definitions of emf. Many authors
∗
(e.g., Hermann Weyl [213]) define emf to be the circulation of E . In that case the emf
is equal to the negative time rate of change of the flux of the laboratory frame magnetic
field B through S. Since the Lorentz force experienced by a charge q moving with the
∗
contour is given by qE = q(E + v × B), this emf is the circulation of Lorentz force
per unit charge along the contour. If the contour is aligned with a conducting circuit,
then in some cases this emf can be given physical interpretation as the work required
to move a charge around the entire circuit through the conductor against the Lorentz
force. Unfortunately the usefulness of this definition of emf is lost if the time or space
rate of change of the fields is so large that no true loop current can be established
(hence Kirchoff’s law cannot be employed). Such a problem must be treated as an
electromagnetic “scattering” problem with consideration given to retardation effects.
Detailed discussions of the physical interpretation of E in the definition of emf are given
∗
by Scanlon [165] and Cullwick [48].
Other authors choose to define emf as the circulation of the electric field in the frame

of the moving contour. In this case the circulation of E in (2.145) is the emf, and is
related to the flux of the magnetic field in the frame of the moving circuit. As pointed
out above, the result differs from that based on the Lorentz force. If we wish, we can
also write this emf in terms of the fields expressed in the laboratory frame. To do this we
must convert ∂B /∂t to the laboratory fields using the rules for a Lorentz transformation.

The result, given by Tai [194], is quite complicated and involves both the magnetic and
electric laboratory-frame fields.
The moving-frame emf as computed from the Lorentz transformation is rarely used as
a working definition of emf, mostly because circuits moving at relativistic velocities are
seldom used by engineers. Unfortunately, more confusion arises for the case v c, since
for a Galilean frame the Lorentz-force and moving-frame emfs become identical. This

is apparent if we use (2.52) to replace B with the laboratory frame field B, and (2.49)

to replace E with the combination of laboratory frame fields E + v × B. Then (2.145)
becomes
d

E · dl = (E + v × B) · dl =− B · dS,
dt S
which is identical to (2.153). For circuits moving with low velocity then, the circulation
of E can be interpreted as work per unit charge. As an added bit of confusion, the term

(v × B) · dl = ∇× (v × B) · dS
S
∗
is sometimes called motional emf, since it is the component of the circulation of E that
is directly attributable to the motion of the circuit.
Although less commonly done, we can also rewrite Ampere’s law (2.142) using (2.151).
This gives
d

H · dl = J · dS + D · dS − (v∇· D) · dS + ∇× (v × D) · dS.
S dt S S S
Using ∇· D = ρ and using Stokes’s theorem on the last term, we obtain

d
(H − v × D) · dl = D · dS + (J − ρv) · dS.
dt S S

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