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Lecture 19-1
Potential Difference Across Inductor
V    I r
+
V
I
internal resistance
• Analogous to a battery
• An ideal inductor has r=0
-
• All dissipative effects are to be
included in the internal resistance (i.e.,
those of the iron core if any)
dI
  IR  L  0
dt
dI
   IR  L  0
dt
Lecture 19-2
Ways to Change Magnetic Flux
 B  BA cos 
• Changing the magnitude of the field within a conducting loop (or coil).
• Changing the area of the loop (or coil) that lies within the magnetic field.
• Changing the relative orientation of the field and the loop.
motor
generator
http://www.wvic.com/how-gen-works.htm
Lecture 19-3
Alternating Current (AC)
= Electric current that changes direction periodically
ac generator is a device which creates an ac emf/current.
A sinusoidally oscillating EMF is induced in a loop of wire
that rotates in a uniform magnetic field.
B  NBA cos  NBA cos t   
dB
 
  NBA  sin t   
dt
where
2
  2 f 
T
ac motor =
ac generator run
in reverse
http://www.wvic.com/how-gen-works.htm
http://www.pbs.org/wgbh/amex/edison/sfeature/acdc.html
Lecture 19-4
Resistive Load
Start by considering simple circuits with
one element (R, C, or L) in addition to
the driving emf. Pick a resistor R first.
+
--
I(t)
Kirchhoff’s Loop Rule:
  t   I t  R  0 ,  t    peak sin t
vR  t   I  t  R   peak sin t
V peak   peak
I t  
Ipeak
Vpeak
R
sin t
vR(t) and I(t) in phase
Lecture 19-5
City lights viewed in a motion blurred exposure. The AC blinking causes
the lines to be dotted rather than continuous (quote from Wikipedia)
Lecture 19-6
Power Dissipated by Resistive Load
VR and I in phase
Power:
P  I R   I peak sin t  R
2
2
  I peak
R  sin 2 t
2
Lecture 19-7
Average Power
P  I 2R
2
Pav  I 2 R  I peak
R sin 2 t
2
 I peak
R sin 2 t
but
sin t  0

 2
1
sin t 

2
1 2 
 Pav   I peak  R
2

Lecture 19-8
Root-Mean-Square Values
I I
2
I rms
 rms
1 2
sin t  I peak
2
2
peak
 2 I rms  1.41 I rms

1 2 
2
 Pav   I peak  R  I rm
sR
2

1
1
  
 peak 
I peak R  I rms R
2
2
2
2
1
 I 
I peak  0.707  I peak
2
I
Similarly,
2
peak
Pav   rms I rms
Lecture 19-9
Non-scored Test Quiz
Which of the following statement is true?
A.
I peak  2 I rms
B.
I peak
1

I rms
2
C.
I peak  2 I rms
D.
1

I rms
2
I peak
V peak
1

Vrms
2
V peak  2Vrms
V peak  2Vrms
V peak
1

Vrms
2
Lecture 19-10
Capacitive Load
Loop Rule:
+
--
q(t )
 (t ) 
0
C
   peak sin  t
q(t )
  peak sin t
C
dq (t )
d  (t )
I (t ) 
C
dt
dt
 C peak cos  t
 v(t ) 
I(t) leads v(t) by 90 (1/4 cycle)


Power: p(t )  I (t )v(t )  I peak cos t VC , peak sin t

VC , peak I peak
2
sin 2t

Pav  0
Lecture 19-11
Inductive Load
Kirchhoff’s Loop Rule:
+
--
dI (t )
 (t )  L
0
dt
 (t )   peak sin t
dI (t )
vL (t )  L
  peak sin t
dt
 peak
I (t )  
cos t
L
vL(t) leads I(t) by 90 (1/4 cycle)


Power: p(t )  I (t )vL (t )   I peak cos t VL , peak sin t

VL, peak I peak
2
sin 2t

Pav  0
Lecture 19-12
Capacitive vs Inductive Load
I(t) leads v(t) by 90
capacitive reactance
1
XC 
C
--
+
Pav  0
VC , peak  X C I peak
vL
vL(t) leads I(t) by 90
inductive reactance
X L  L
VL , peak  X L I peak
+
--
Lecture 19-13
(Ideal) LC Circuit
• From Kirchhoff’s Loop Rule
Q
dI
L 0
C
dt
• From Energy Conservation
2
Q 2 1 2  Q peak
E
 LI  
2C 2
 2C
dQ
I
dt
dE
0
dt
same

  const.

Q dQ
dI
 LI
0
C dt
dt
d 2Q  1 

Q  0
2
dt
 LC 
Q  Q peak cos(0t   )
dQ
I
 0Q peak sin( 0t   )
dt
Q
dI
L 0
C
dt
harmonic oscillator with angular
1
frequency
0 
LC
Natural Frequency
Lecture 19-14
LC Oscillations
Q2
1 2
dQ
UE 
, U B  LI , I 
2C
2
dt
No Resistance =
No dissipation
Lecture 19-15
Physics 241 –Quiz 16b – March 20, 2008
In most of Europe, the peak voltage of household
outlets is 311 V. What is the rms voltage?
a) 110 V
b) 141 V
c) 156 V
d) 220 V
e) 311 V
Lecture 19-16
Physics 241 –Quiz 16c – March 20, 2008
In Japan, the rms voltage of household outlets is
100V. What is the peak voltage?
a) 200 V
b) 141 V
c) 100 V
d) 50 V
e) 71 V