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15 Inductance — solenoid, shorted coax
•
Given a current conducting path
C
,themagnet
icfux
Ψ
linking
C
can
be expressed as a Function oF current
I
circulating around
C
.
Ψ
I
I,
E
=

L
dI
dt
V
(
t
)=
L
dI
dt
+

±
3
±
2
±
1
0
1
2
3
±
3
±
2
±
1
0
1
2
3
x
z
•
IF the Function is linear, i.e., iF we have a
linear fuxcurrent relation
Ψ=
LI,
then constant
L
=
Ψ
I
is termed the
selfinductance
1
oF path
C
,ane
lementary
inductor
.
–
Di±erentiating the fuxcurrent relation with respect to time
t
,and
using the Fact that
E
=

d
Ψ
dt
,
we ²nd that the emF oF inductor
L
is simply
E
=

L
dI
dt
,
which is a voltage rise across the inductor in the direction oF cur
rent
I
(which, oF course makes
L
dI
dt
avo
l
taged
ropinthesame
direction, as used in circuit courses).
1
Amu
tua
linduc
tance
M
12
,bycon
t
ra
s
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For an inductor consisting of
n
loops, the inductance
L
≡
n
Ψ
I
,
and the emf
E
measured around
n
loops is
E
=

d
dt
n
Ψ=

L
dI
dt
once again.
Example 1:
An
n
turn coil has a resistance
R
=1Ω
and inductance of
1
μ
H. If it is
conducting 3 A of current at
t
=0
,determ
ine
I
(
t
)
for
t>
0
.
Solution:
Current Fow in the resistive
n
turn coil will be driven by emf
E
=

L
dI
dt
matching the voltage drop
RI
.H
en
ce

L
dI
dt
=
RI
↔
dI
dt
+
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 Spring '08
 Kim

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