14 Faraday’s law and induced emf
Michael Faraday discovered (in 1831, less than 200 years ago) that a
changing
current in a wire loop
induces
current flows in nearby wires — today we
describe this phenomenon as
electromagnetic induction
: current change
in the first loop causes the magnetic field produced by the current to change,
and magnetic field change, in turn,
induces
1
(i.e., produces) electric fields
which drive the currents in nearby wires.
Definitions of
E
and
B
have not changed:
recall that
•
E
is force per unit sta
tionary charge
•
B
gives an additional
force
v
×
B
per
unit
charge in motion with
velocity
v
in the mea
surement frame.
•
While static electric fields produced by static charge distributions are
unconditionally curlfree,
timevarying electric fields
produced by cur
rent distributions with timevarying components are found to have, in
accordance with Faraday’s observations, nonzero curls specified by
∇ ×
E
=

∂
B
∂
t
Faraday’s law
at all positions
r
in all reference frames of measurement. Using
Stoke’s
theorem
, the same constraint can also be expressed in
integral form
as
C
E
·
d
l
=

S
∂
B
∂
t
·
d
S
Faraday’s law
for all surfaces
S
bounded by all closed paths
C
(with the directions of
C
and
d
S
related by right hand rule).
1
Relativistic derivation of static
B
given in Lecture 12 can be extended to show that Coulomb interactions of charges
in
timevarying
motions require a description in terms of timevarying
B
and
E
— see, e.g.,
Am. J. Phys.
: Tessman, 34,
1048 (1966); Tessman and Finnel,
35
, 523 (1967); Kobe,
54
, 631 (1986). Thus, the
cause
of
induced
E
is not really the
timevarying
B
, but rather the timevarying current
J
producing the variation in
B
. Still, we find it convenient to attribute
the induced
E
to timevarying
B
mainly because Ampere’s law provides an explicit link of a timevarying
J
to
B
(see Lect’s
12 and 16).
1
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•
The right hand side of the integral form equation above includes the
flux of
rate of change of magnetic field
B
over surface
S
. If
contour
C
bounding
S
is stationary in the measurement frame, then
the equation can also be expressed as
C
E
·
d
l
=

d
dt
S
B
·
d
S
,
where the right hand side includes the
rate of change
of
magnetic flux
Ψ
≡
S
B
·
d
S
that is independent of
S
(so long as bounded by
C
, by Stoke’s theorem).
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 Spring '08
 Kim
 Electromagnet, Electromotive Force, Magnetic Field, Faraday's law of induction, dΨ

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