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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Lecture 11
Faraday’s Law and Electromagnetic Induction
and
Electromagnetic Energy and Power Flow
In this lecture you will learn:
• More about Faraday’s Law and Electromagnetic Induction
• The Nonuniqueness of Voltages in Magnetoquasistatics
• Electromagnetic Energy and Power Flow
• Electromagnetic Energy Stored in Capacitors and Inductors
• Appendix (some proofs)
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Faraday’s Law Revisited
a
d
H
t
a
d
B
t
s
d
E
o
r
r
r
r
r
r
.
.
.
∫∫
∂
∂
−
=
∫∫
∂
∂
−
=
∫
µ
A closed contour
B
Faraday’s Law
: The line integral of E
field over a closed contour is equal to
–ve of the time rate of change of the
magnetic flux that goes through any
arbitrary surface that is bounded by
the closed contour
Important Note:
In
electroquasistatics
the line integral of Efield over a closed
contour was always zero
In
magnetoquasistatics
this is NOT the case
( )
∫∫
=
∇
−
=
0
.
.
s
d
s
d
E
r
r
r
φ
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Electromagnetic Induction and Kirchoff’s Voltage Law
Now consider a circuit through which the magnetic
flux is changing with time (Kirchoff’s voltage law is
violated)
+
+


dt
d
a
d
B
t
s
d
E
λ
−
=
∫∫
∂
∂
−
=
∫
⇒
r
r
r
r
.
.
dt
d
V
R
I
R
I
s
d
E
−
=
−
+
=
∫
⇒
2
1
.
r
r
1
R
2
R
I
()
dt
d
R
R
R
R
V
I
2
1
2
1
1
+
−
+
=
⇒
V
+

( )
t
+
+


1
R
2
R
I
V
+

Kirchoff’s voltage law comes from the electroquasistatic
approximation:
0
.
=
∫
s
d
E
r
r
0
.
2
1
=
−
+
=
∫
⇒
V
R
I
R
I
s
d
E
r
r
2
1
R
R
V
I
+
=
⇒
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Lenz Law
+
+


Suppose an induced current
I
is flowing through the wire:
dt
d
R
R
I
2
1
1
+
−
=
⇒
1
R
2
R
I
The induced current in the wire produces its own
magnetic field
Lenz Law
is just an easy way to remember in which direction the induced current flows
The law states that the induced current will flow in a direction such that its own
magnetic field opposes the time variation of the magnetic field that produced it
Example:
Suppose the magnetic flux through the wire loop shown above was
increasing with time (so that
d
/
dt
> 0).
Lenz would tell us that the induced current would flow in the clockwise direction so
that its own magnetic field would oppose the increasing magnetic flux through the loop
In the equation above this fact comes out from the negative sign on the right hand side
( )
t
3
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
NonUniqueness of Voltages in Magnetoquasistatics  I
+
+


()
dt
d
R
R
I
s
d
E
λ
−
=
+
=
∫
2
1
.
r
r
1
R
2
R
I
2
V
1
V
Question:
What is the voltages difference
V
2

V
1
?
One may be tempted to write …….
1
1
2
2
2
1
V
R
I
V
V
R
I
V
=
−
=
−
0
0
1
2
=
−
⇒
=
⇒
V
V
I
….. which cannot be correct since we know that:
We have:
What went wrong?
Our usual concepts of circuit theory and potentials which are
based on conservative Efields are not valid when time varying magnetic fields
are present
dt
d
R
R
I
2
1
1
+
−
=
( )
t
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
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This note was uploaded on 02/02/2008 for the course ECE 3030 taught by Professor Rana during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 RANA
 Electromagnet, Volt

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