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Notes 4
Spring 2005.doc
1
Faraday’s Law of Induction and the Electromotive Force
Early in the history of the scientific examination of electromagnetism it was noted experimentally that
a voltage can be induced in a loop of wire when a magnetic flux passing through the interior of the
wire loop changes with time.
This induced voltage subsequently drives a current in the wire.
Faraday’s law of induction was written to describe this phenomenon:
dt
d
emf
Φ
−
=
V
In this equation the
is the voltage (
e
lectro
m
otive
f
orce, in volts) that is ‘induced’ by the time
changing magnetic flux,
, through the closed path.
The minus sign indicates the sense of the voltage
polarity as it relates to the changing flux.
emf
V
Φ
It is important to note that with or without a wire present a changing magnetic flux will induce a
voltage along the path defining any area through which the flux passes.
However, current will only
flow if the path is conducting.
This induced voltage (
) is equally described by the path integral of the electric field around the
path of interest:
emf
V
emf
d
V
l
E
=
∫
⋅
G
G
where
is the electric field strength along the closed path and
E
G
l
d
G
is a differential line element along
that path.
Dimensional analysis of this equation shows that the result of this line integral is indeed a
voltage.
We then write
emf
dt
d
d
V
l
E
=
Φ
−
=
⋅
∫
G
G
Note that in determining the magnetic flux
Φ
one has the option of choosing one of two directions
for
S
d
G
and in executing the path integral one has a choice in direction as well.
Since
∫
⋅
=
Φ
S
S
d
G
G
B
, the
choice of direction for
S
d
G
will determine the algebraic sign for
Φ
and subsequently the algebraic sign
of
dt
d
Φ
and finally, the sign on the induced emf.
Therefore, the opposing choices for
S
d
G
have the
effect of changing the algebraic sign on the answer for the induced emf.
We must, however, demand
that the physical reality is unambiguously determined through the mathematics and there can only be
one reality for a given configuration.
To extract the correct “reflection of reality”, when we perform the closed path integral
∫
⋅
l
d
G
G
E
we
must move in a direction corresponding (through the righthand rule) to our choice of direction for
S
d
G
in order to make the above boxed equation legitimate.
We use the right hand rule with our thumb in
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Spring 2005.doc
2
the direction of
S
d
G
and our fingers curling in the direction we must move in executing the path
integral.
This formulation forces the algebraic signs of the emf on both sides of the above equation to be
consistent with each other.
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This note was uploaded on 01/20/2012 for the course EE 4460 taught by Professor Czarnecki during the Fall '10 term at LSU.
 Fall '10
 Czarnecki
 Electromagnet, Electromotive Force, Flux, Volt

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