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9. Inductance
1) Mutual inductance
1
2
I1
I1  B2   2
 2  N 21  M 21I1
1  B1  I 2
1  N11  M 12 I 2
 B
 
t
I 1
 2  M
t
M12  M 21  M
M is a geometrical factor!
I 2
1  M
t
2) Self-inductance
1
  N  LI
Units: (henry)
  L
I
t
L is a geometrical factor!
L  M   1H  1Wb / A  1V  s / A  1  s  1J / A2
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3) Example: Self-inductance of a solenoid
Three geometrical quantities describe a solenoid:
A – area of cross-section
l – length (or V=Al – volume)
N – number of turns (or n=N/l – number of turns per unit length)
Self-inductance L should be a function of these quantities,
and should be independent from the current in the solenoid!
  N1  LI
L
 1  BA  0 nIA
N 1
 1
I
is proportional to I !
L  0 n 2 Al  0 n 2V  0 N 2 A / l
Example:
r  2cm
l  20cm
N  200
L ?
4 10
L
7


T  m / A 200  2 10  2 m
20 10  2 m
2

2
 32 2 10 6 H
L  320 H
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10. RL circuits
1) Circuit with emf
   ind  IR
 L
I
R
I
 IR
t

I R

 I 0
t L
L
I


1 e 
R
t / 
I max 

R
L
 ind   L
I
t
  L/ R
Relaxation time
2) Circuit without emf
 ind  IR
I R
 I 0
t L
e  2.718
I  I 0 e  t /
e 1  0.368
1  e 1  0.632
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Example: Two tightly wound solenoid have the same length and circular crosssectional area. They use wires made from the same material, but solenoid 1 uses
wire that is half as thick as solenoid 2. What is the ratio of their inductances? What
is the ratio of their time constants (assuming no other resistance in the circuits)?
l1  l2  l
a)
A1  A2  A
L2   n Al
2
0 2
1   2  
r1  r2 / 2
n1  2n2
lw1  2lw 2
Aw1  Aw 2 / 4
a ) L1 / L2  ?
b) 1 /  2  ?
L1  0 n12 Al
b)
R1  
lw1
Aw1
lw 2
R2  
Aw 2
 1  L1 / R1
 2  L2 / R2
L1 / L2  n / n  n1 / n2   4
2
1
2
2
2
R1 / R2 
lw1 / lw 2
2

8
Aw1 / Aw 2 1 / 4
1 / 2 
L1 / L2 4 1
 
R1 / R2 8 2
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11. Energy stored in an inductor
I
 L
 IR 
t
I
I  LI
 I 2R
t
power from
the battery
U
I
 LI
t
t
dissipation in
the resister
power supplied
to the inductor
U  12 LI 2
For any inductor!
11a. Energy of magnetic field
For solenoid:
L  0 n 2V
B 2V
U  LI  0 n V B / 0 n  
20
1
2
B  0 nI  I  B / 0 n
Magnetic energy density:
(for any magnetic field B)
Example:
B  0.20T
u ?
U
B2
u 
V 20
2
1
2
2
2
V – volume
0.20T 2
105
4
3
u

T

A
/
m

1
.
6

10
J
/
m
2  4 10 7 T  m / A 2
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12. AC circuits and reactance
1) Resistor
I  I 0 cos t
V  IR  I 0 R cos t  V0 cos t
I rms  I 0
Vrms  V0
t
  2f 
2
2
Current & voltage
are in phase
2
2
P  IV  I rms
R  Vrms
R
2
T
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2) Inductor
I  I 0 cos t
V L
I
0
t
V   LI 0 sin t  V0 cos(t  90 )
V0  LI 0  I 0 X L
The current lags
the voltage by 90°
Vrms  LI rms  I rms X L
V
I
t
inductive reactance:
X L  L  2fL
If the frequency is low then the reactance is small!
If the frequency is high then the reactance is big!
Example:
L  0.30 H
Vrms  120V
f a  60 Hz
f b  600 Hz
I rms  ?
X La  2f a L  2 60 Hz   0.30 H   113
X Lb  2f b L  2 600 Hz   0.30 H 0.30 H   600 Hz  1130
I rms( a )  Vrms / X La  120V  / 113  1.06 A
I rms(b )  Vrms / X Lb  120V  / 1130  0.106 A
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3) Capacitor
I  I 0 cos t
Q I0

sin t  V0 cos(t  90 )
C C
I
The current leads
V0  0  I 0 X C
the voltage by 90°
L
I
Vrms  rms  I rms X C
L
1
1

capacitive reactance: X C 
C 2fC
V
I  Q / t
t
I
V
If the frequency is low then the reactance is big
If the frequency is high then the reactance is small!
Example:
C  1.0 F
Vrms  120V
XC 
I rms  Vrms / X C  2fCVrms
1
2fC


f a  60 Hz
I rms( a )  2 60 Hz  1.0  10 6 F 120V    1.44  10 2 A
f b  600 Hz
I rms( a )  0.045 A
I rms  ?
I rms( b )  0.45 A
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