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Unit 5 Induction/p211_05e

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Self Inductance
Solenoid Flux
 Coils of wire carrying current
NI,l
generate a magnetic field.
• Strong field inside solenoid
B
 If the current increases then
F
the magnetic field increases.
• Increased magnetic flux
 The magnetic flux through all
B 
 0 NI
l
coils depends on the coil area
and the number of turns.
0N A
2
F  NAB 
l
I
Inductance
 The ratio of magnetic flux to current is the inductance.
L
F
I
 Inductance is measured in henrys.
• 1 H = 1 T m2 / A
• More common, 1 H = 1 V / A / s
 The inductance can be derived for an ideal solenoid.
0N A
2
L
l
 0 N r
2

l
2
Coil Length
 An inductor is made by
wrapping a single layer of wire
around a 4.0-mm diameter
cylinder. The wire is 0.30 mm
in diameter.
 What coil length is needed to
have an inductance of 10 H?
 The formula is based on the
radius and length of the coil.
 0 N r
2
L
2
l
• Radius r = 2.0 x 10-3 m
• Turns N = l / d
 Substitute for N in the formula.
 0 (l / d )  r
2
L
l
2

 0 l r
d
2
l
d L
 0 r
2
 0 . 057 m
2
2
Electric Inertia
 A changing current will
Increasing I, F
create a changing flux.
B
 Faraday’s law states that the
changing flux will create an
emf.
• Direction from Lenz’s law
 The emf acts to oppose the



Decreasing I, F
change in flux.
B
• Inertial response



Back-Emf
 Motors have internal coils.
 The self-inductance will
oppose a change in current by
creating an emf.
 This back-emf is responsible
for excess power draw when a
motor starts.
Induced EMF
 Faraday’s law gives the
magnitude of the induced
emf.
 
F
M
t
• Depends on rate of change
 The definition of inductance
gives a relationship between
voltage and current.
• More useful in circuits
 Inductive elements in a
circuit act like batteries.
• Stabilizes current
  L
I
t
Mutual Inductance
 The definition of inductance
applies to transformers.
• Mutual inductance vs selfinductance
VA
R
N
A
VB   N B
NB
F
t
  NB
F
M
t
M
I
t
 Mutual inductance applies to
both windings.
M
Stored Energy
 Electrical power is voltage times current.
• True for emf from inductance
• Average current is approximately one half maximum
• Use one half to get average power
Pav   I av  L
I
t
1
I av 
L
2
I
t
I
 Magnetic energy is stored in a magnetic field.
• Energy is the power times the time
U  Pav  t 
1
2
LI
2
Energy Density
 The energy density in a
 The energy density in a
solenoid is based on its
volume.
uB 
U
r l
2

1
2
0n I
2
capacitor is based on its
volume.
2
uE 
 The energy density can be
uB 
1
2
0B
CV
Ad
2

1 K 0
2 d
2
V
2
 The energy density can be
expressed in terms of the
magnetic field.
B   0 nI
1
2
expressed in terms of the
electric field.
2
E 
V
d
uE 
1
2
K 0E
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