DC circuits
Physics 122
10/16/13
Lecture X
1
Skills
•  Solve problems on parallel and series
connections in DC circuits
•  RC circuits
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Lecture X
2
1
Symbols
•  Circuits can be
rearranged:
–  Wires with negligible
resistance can be
–  Stretched
–  Bended
–  Collapsed to a point
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Lecture X
3
Electric battery
•
•
•
•
For idealized battery emf – supplied voltage
Real life batteries have internal resistance r
Terminal voltage (supplied to the circuit)
Vab=emf-Ir
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Lecture X
4
2
Skiing ßà electric circuit
a
i li
ft
b
Batt
ery
Sk
c
Cannot stop at b, must get to c –
ski lift:
V=V1+V2 - Net voltage drop in a
circuit is always equal to the
supplied voltage (e.g. battery)
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Lecture X
5
Series connection
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•  Charge conservation:
–  I=I1=I2=I3
•  Ohm’s law
–  V1=IR1; V2=IR2; V3=IR3
•  Energy conservation:
–  qV=qV1+qV2+qV3
–  V=V1+V2+V3
•  IReq=IR1+IR2+IR3
•  Req=R1+R2+R3
Lecture X
6
3
Parallel connection
1
1 1
1
= + +
Req R1 R2 R3
•  Charge conservation: I=I1+I2+I3
•  Energy conservation: V=V1=V2=V3
•  Ohm’s law: I1=V/R1; I2=V/R2; I3=V/R3
V
V V V
= + +
Req R1 R2 R3
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Lecture X
7
Capacitor connections
No DC current
through capacitor,
Just store charge.
•  Parallel connection:
•
•  Same voltage – V1=V2=V3=V •
•  Different charges:
•
–  Q1=C1V Q2=C2V
Q3=C3V
–  Q=Q1+Q2+Q3=V(C1+C2+C3)
Series connection:
Same charge (QA=0 à -Q1+Q2=0)
Different voltages V=V1+V2+V3
Q
Q Q Q
= +
+
Ceq C1 C2 C3
–  Ceq=C1+C2+C3
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Lecture X
1
1
1
1
= +
+
Ceq C1 C2 C3
8
4
RC circuit
•  RC circuit contains at least one capacitor (C ) and at
least one resistor (R ).
I=
dQ
dt
VR = IR
VC = Q / C
VR + VC = Vbattery = const
+Q
dQ Q
R
+ =V
dt C
RCQ '+Q = CV
C
-Q
Q = CV (1 − e −t /τ )
τ = RC
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Lecture X
9
DC circuits
•
•
•
•
Series connection
I=I1=I2=I3
V=V1+V2+V3
Req=R1+R2+R3
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•
•
•
•
Parallel connection
I=I1+I2+I3
V=V1=V2=V3
1
1 1
1
= + +
Req R1 R2 R3
Lecture X
10
5
Series vs parallel - Req
•  R1=R2=R3=R
•  Req=3R
•  R1=R2=R3=R
•  Req=R/3
>
Equivalent resistance is larger in series connection.
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Lecture X
11
Series vs parallel - I
•
•
•
•
R1=R2=R3=R
Req=3R
I=V/(3R)
I1=I2=I3=I=V/(3R)
<
<
•
•
•
•
R1=R2=R3=R
Req=R/3
I=3V/R
I1=I2=I3=I/3=V/R
Total current and individual currents are smaller in series connection.
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Lecture X
12
6
Series vs parallel - P
P1=I2R
Pnet=IV
Brightness
proportional
to power
•
•
•
•
R1=R2=R3=R
Req=3R
I=V/3R à Pnet=V2/3R
I1=V/3Rà P1=V2/9R
<
<
•
•
•
•
R1=R2=R3=R
Req=R/3
I=3V/R à Pnet=3V2/R
I1=V/R à P1=V2/R
Total and individual power consumptions are smaller in series connection.
Light bulbs are brighter in parallel connection.
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Lecture X
13
Safety and grounding
•  Usually V is fixed
•  To reduce I – increase R
•  Dry surface
–  1mA – “feel”
•  Insulating sole
–  10mA – muscle contraction
•  Insulating floor
–  70 mA – “ventricular fibrillation”
–  ~1A – hearts stops (not as harmful)•  To avoid high V •  Ground the case
–  >10A – “fry”
(3prong plug) 14
10/16/13
Lecture X
•  Voltage does not kill.
•  Current kills
7
Kirchhoff’s rules
•  Junction rule (charge conservation):
–  At any junction point the sum of all current entering the
junction must equal the sum of all currents leaving the
junction
•  Do not disregard signs!
•  Assign initial signs based on your best guess – if you were wrong the
currents will turn to be negative (opposite to initial guess)
•  Loop rule (energy conservation):
–  The sum of the changes in potential around any closed path of
a circuit must be zero.
•
•
•
•
•
Choose a direction in which you circle the loop (random)
For resistor Vab=-IR – negative if current along the loop direction
For resistor Vab=+IR – positive if current opposite the loop direction
For battery Vab – positive if the direction is from – to +
For battery Vab –negative if the direction is from + to -
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Lecture X
15
Series vs parallel - Req
•  R1=Rà2R
•  Req=3Rà4R
R2=R3=R
•  R1=Rà2R
R2=R3=R
•  Req=R/3=2R/6à2R/5
If any of R’s increased the equivalent resistance will increase
in both parallel and series connection.
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Lecture X
16
8
Series vs parallel - I
•
•
•
•
R1=Rà2R
R2=R3=R
Req=3Rà4R
I=V/Req=V/3RàV/4R
I1=I2=I3=I
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•
•
•
•
•
R1=Rà2R
R2=R3=R
Req=R/3=2R/6à2R/5
I=V/Req=6V/2Rà5V/2R
I1=V/R1=V/RàV/2R
I2=I3=V/R – the same
Lecture X
17
Test problem
•  For resistors connected in series, which of the
following will be the same for each of the
resistors?
•  A
The current through.
•  B
The potential difference across.
•  C
The resistance of.
•  D
The power dissipated by.
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Lecture X
18
9
Test problem
•  For resistors connected in parallel, which of the
following will be the same for each of the
resistors?
•  A
The current through.
•  B
The potential difference across.
•  C
The resistance of.
•  D
The power dissipated by.
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Lecture X
19
10