physics for scientists and engineers ii, summer semester 2009 1 lecture 18: july 6 th 2009 physics...
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Physics for Scientists and Engineers II , Summer Semester 2009
1
Lecture 18: July 6th 2009
Physics for Scientists and Engineers II
Physics for Scientists and Engineers II , Summer Semester 2009
2
Energy loss in the circuit
LRt
LRt
LRt
etRI
edt
dILILI
dt
d
edt
dQ
C
Q
C
Q
dt
d
)(dt
dU
resistor) (in theheat tosystemin energy of Loss
2
1
dt
dU :Inductor
2
1
dt
dU :Capacitor
:components in theenergy of change of Rate
2R
2L
2c
Physics for Scientists and Engineers II , Summer Semester 2009
3
Energy loss in the circuit
tttI
t
ddd
d
sineQcose2L
RQ)(
coseQQ(t)
2LRt-
max2L
Rt-max
2LRt-
max
teC
Q
C
tQt L
Rt2
2max
2
c cos22
1 U:Capacitorin Energy
tt
L
Rtt
L
RQLe
tILt
ddd
dddL
Rt sincos
2sincos
42
1
2
1 U:Inductorin Energy
2222
22max
2L
dtttL
Rtt
L
RQt
dttIRtt
ddd
dddL
Rt
0
2222
22maxR
2R
sincos2
sincos4
ReU
dU :resistorin heat toconvertedEnergy
Physics for Scientists and Engineers II , Summer Semester 2009
4
Example from book: Chapter 32, Problem 18
00.4
V10 00.4
00.8
H00.1
r. thereafte timeoffunction
a asswitch in thecurrent theandinductor in thecurrent Find
0.at t closed is time)long afor open (which wasswitch The
Physics for Scientists and Engineers II , Summer Semester 2009
5
Use Kirchhoff’s Rules
00.4
V10 00.4
00.8
H00.11 2
1I2I
3I
)3(000.100.800.4 :2 Loop
)2(000.400.4-10V :1 Loop
)1( :RuleJunction
223
31
321
dt
dIHII
II
III
Physics for Scientists and Engineers II , Summer Semester 2009
6
Solving the system of equations
)3(000.100.800.4 :2 Loop
)2(000.400.4-10V :1 Loop
)1( :RuleJunction
223
31
321
dt
dIHII
II
III
)4(000.8I00.4I-10V
000.4I00.4)I(I-10V :(2) into )1(
32
332
)4(000.20.20I-10V
000.200.16I00.4I-10V :(3) 2 )4(
22
222
dt
dIH
dt
dIH
Physics for Scientists and Engineers II , Summer Semester 2009
7
Solving the system of equations
)4(000.20.20I-10V 22
dt
dIH
22 I0.20I20.0-10V x:onSubstituti ddx
tH
x
x
t
exx
tHx
xdt
Hx
dx
dtHx
dx
dt
dxHx
0.10
0
0 0
0.10ln0.10
0.1000.20
00.2
0
tH
tH ee
0.100.10
2
2
10V0I20.0-10VI20.0-10V
I20.0-10V x:againon Substituti
Physics for Scientists and Engineers II , Summer Semester 2009
8
Solving the system of equations
tH
tH
tH
eAe
e
0.100.10
2
0.10
2
15.020.0
10V-10VI
10VI20.0-10V
tH
tH
tH
AA
VV
I10.0-
10.0-
3
3
10.0-
32
e-125.025.100.8
e-100.20.10
000.8I00.4e-10.5A-10V
)4(000.8I00.4I-10V
Eq.(4) intoResult
Physics for Scientists and Engineers II , Summer Semester 2009
9
Solving the system of equations
t
HeA0.10
2 15.0I
t
HAAI10.0-
3 e-125.025.1
tH
tH
tH
AA
AAAIII
10.0-
10.0-10.0-
321
e-125.025.1
e-125.025.1e-15.0
Physics for Scientists and Engineers II , Summer Semester 2009
10
Solving the system of equations
t
HeA0.10
2 15.0I
t
HAAI10.0-
3 e-125.025.1
t
HAAI10.0-
1 e-125.025.1
A5.0I2
AI 5.11
AI 0.13
For long times t:
00.4
V10 00.4
00.8 67.2
67.6totalR
AV
I 5.167.6
101
Physics for Scientists and Engineers II , Summer Semester 2009
11
AC (“alternating current”) Sources
R
tVv sinmax
msf
T 7.161
srad377 60Hzf : USin they electricit Household
Rv
-2
0
vmaxV
T
Physics for Scientists and Engineers II , Summer Semester 2009
12
Resistors in AC Circuits
R
tVv sinmax
tItR
V
R
ΔviRi-Δv RR sinsin0 :rule loop sKirchhoff' max
max
Rv
Ri
tRIRiΔv RR sin :resistor across drop Voltage max
resistorrough Current th
-1.3
0
Current
Voltage
maxI
maxV
Physics for Scientists and Engineers II , Summer Semester 2009
13
Resistors in AC Circuits
tIiR sinmax
tVΔvR sinmax
-1.3
0
Current
Voltage
maxI
maxV
Applied voltage and resulting current reach their peaks at the same time They are “in phase”.
Physics for Scientists and Engineers II , Summer Semester 2009
14
Phasor Diagram
-1.3
0
Current
Voltage
maxI
maxV
In phasor diagram: Vectors rotate counterclockwise with frequency . The projection of the rotating vectors onto the vertical axis yields the instantaneous values of voltage and current.
RR vi ,
maxI
maxV
t
Physics for Scientists and Engineers II , Summer Semester 2009
15
Average current, rms current, average power
Current
-4
00 iR
Current Squared
-4
0
2max
2R 2
1 i I
i2R
Ri
2Ri
maxmax
maxmax
2
2
max2max
22
707.02
707.02
22
1
:Rin dissipatedpower average The
VV
V
II
I
RIP
RI
RI
Ri RiP
rms
rms
rms
RR
Physics for Scientists and Engineers II , Summer Semester 2009
16
Measuring AC voltage and AC current with volt/current meters.
02
V indicate
will voltmeter themode, AC theusing measuredproperly If
present. is voltage
no that believe you to leading zero, be willread valueThe
:mode DCin voltmetera using voltageACan measureyou If
:Careful
.I / V read meter will themode, AC""In
.I / V read meter will themode, DC""In
:AC and DCfor settings separate have meterscurrent Volt /
maxrms
rmsrms
averageaverage
V
Physics for Scientists and Engineers II , Summer Semester 2009
17
-1.3
0
Current
Voltage
Inductors in an AC Circuits
L
tVv sinmax
2sin
cossin
sin
sin0 :rule loop sKirchhoff'
max
maxmax
max
max
tL
Vi
tL
Vdtt
L
Vi
dttL
Vdi
dt
diLtV
dt
diL-Δv
L
L
L
LL
Lv
Li
maxI
maxV
Physics for Scientists and Engineers II , Summer Semester 2009
18
Inductors in an AC Circuits
-1.3
0
Current
Voltage
LL vi ,
maxI
maxV
t
For sinusoidal voltages:The phase of the current through the inductor lags behind the voltageacross the inductor by 90 degrees (one quarter cycle in time).
Lv
Li
Physics for Scientists and Engineers II , Summer Semester 2009
19
Inductors in an AC Circuits
ω)). toaccording (reacts ωon
depends that form a hasXfact that thefrom comes reactance term(the
.reactance" inductive" called is LωXquantity The
e).(resistanc of units has
resistor. afor to thisCompare
L
L
maxmax
maxmax
L
R
VI
L
VI
0IlimXlim
small). is(current large very is X voltages,changingrapidly For very
Ilim0Xlim
large). is(current small very is X voltages,changingslowly For very
:implies LωX
maxω
Lω
L
max0ω
L0ω
L
L
LX
V
L
VI maxmax
max
Physics for Scientists and Engineers II , Summer Semester 2009
20
-1.3
0
Current
Voltage
Capacitors in an AC Circuits
C
tVv sinmax
2sin
cos
sin
sin0 :rule loop sKirchhoff'
max
max
max
max
tVCi
tVCdt
dqi
dttVCqC
qtV
C
q-Δv
C
C
Cv
Ci
maxI
maxV
Physics for Scientists and Engineers II , Summer Semester 2009
21
-1.3
0
Current
Voltage
Capacitors in an AC Circuits
CC vi ,
maxI
maxV
t
For sinusoidal voltages:The phase of the current to the capacitor leads the voltageacross the capacitor by 90 degrees (one quarter cycle in time).
CvCi
Physics for Scientists and Engineers II , Summer Semester 2009
22
Capacitors in an AC Circuits
.reactance" capacitive" called is 1
Xquantity The
e).(resistanc of units has1
resistor. afor to thisCompare
C
maxmaxmaxmax
C
C
R
VIVCI
maxω
Cω
C
max0ω
C0ω
C
C
Ilim0Xlim
large). is(current small very is X voltages,changingrapidly For very
0IlimXlim
small). is(current large very is X voltages,changingslowly For very
:implies 1
X C
CX
VVCI max
maxmax