PHYS 272: Electric and Magnetic Interactions
Circuit Elements
Jonathan Nistor
Tuesday, July 16th, 2013
Jonathan Nistor (Purdue University)
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Lecture 20
Circuit Elements
20.4 Batteries
20.5 Ammeters, Voltmeters, Ohmeters
20.6 Quantitative Analysis of RC Circuits
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Work and Power in a Circuit
Current: charges are moving  work is done
Work = change in electric potential energy of charges
W=
U e  q  V
Power = work per unit time:
P
U e q  V q


V
t
t
t
I
Power for any kind of circuit component: P = IΔV
Units: AV 
Jonathan Nistor (Purdue University)
CJ J
 W
sC s
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Power Dissipated by a Resistor
Know V, find P
emf
R
Know V, find P
In practice: need to know P to select right size resistor – capable
of dissipating thermal energy created by current.
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Lecture 20
Real Batteries: Internal Resistance
Model of a real battery
“Ideal battery” plus internal
resistance. Assume this
resistance to be ohmic!
R
Internal resistance affects the behavior of a circuit!
Round-trip (energy conservation)
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Ammeters, Voltmeters, and Ohmmeteres
Ammeter: measures current I
Voltmeter: measures voltage difference ΔV
Ohmmeter: measures resistance R
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Using an Ammeter
When connecting the Ammeter,
conventional current must flow into the ‘+’
terminal and emerge from the ‘–’ terminal
to result in a positive reading
0.150
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An ammeter must be inserted in series
with the circuit element in which current is
being measured! Why?
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Ammeter Design
Simple ammeter using your lab kit:
Simple commercial
ammeter
Digital ammeters: use semiconductor elements.
ADC – analog-to-digital converter
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Ammeter Design – internal resistance
A
rint
emf
R
Ammeter is inserted in series in a circuit:
Measures current flowing through it!
The “act of measuring” requires charges to do some
work: Internal Resistance
No ammeter:
emf  RI  0
With ammeter: emf  rint I  RI  0
I
emf
R
emf
I
R  rint
Internal resistance of an ammeter must be very small
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Using a Voltmeter
When connecting the Voltmeter, higher
potential must be connected to the ‘+’
socket and lower potential to the ‘–’
socket to result in a positive reading
A voltmeter is basically an ammeter in
series with a resistor of known
resistance!
A voltmeter must be connected in
parallel with the circuit element to be
measured, why?
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Lecture 20
Voltmeter design – Internal Resistance
A voltmeter must be connected in
parallel with the circuit element to be
measured, why?
R1
emf
B
rint
R2
A
A
VAB in absence of a voltmeter:
R2
VAB 
emf
R1  R2
VAB in presence of a voltmeter:
R2||int
VAB 
emf
R1  R2||int
R2 rint
R2||int 
R2  rint
Internal resistance of an voltmeter must be very large!
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Ohmmeter
Ohmmeter: Ammeter with a small
voltage source
A
R
Knows ΔV , measures I, calculates R
Ohmmeter
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RC Circuits – Quantitative Analysis
Vround_ trip  emf  RI  VC  0
Q
0
C
dQ emf  Q / C
I

dt
R
emf
I0 
R
emf  RI 
Initial situation: Q=0
Q and I are changing in time
dI d  emf  d  Q 
 
 

dt dt  R  dt  RC 
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Q
VC 
C
d
dt
dI
1 dQ

dt
RC dt
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dI
1

I
dt
RC
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RC Circuits – Current
dI
1

I
dt
RC
1
1
dI  
dt
I
RC
I
t
1
1
dI


dt
I I

RC 0
0
t
ln I  ln I 0  
RC
I
t
ln  
I0
RC
I
e
I0

Current in an RC circuit:
I  I 0et / RC
What is I0 ?
Current in an RC circuit:
emf t / RC
I
e
R
t
RC
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RC Circuits – Charge and Voltage
What about charge Q?
I
dQ
dt
dQ  Idt
Current in an RC circuit:
t
I  I 0et / RC
emf
Q   Idt 
R
0
t
t / RC
e
dt

0
Q  C emf 1  et / RC 
Current in an RC circuit:
emf t / RC
I
e
R
V 
Q
C
Check: t =0, Q=0, t inf, Q=C*emf
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RC Circuits – Summary
Current in an RC circuit
emf t / RC
I
e
R
Charge in an RC circuit
Q  C emf 1  et / RC 
Voltage in an RC circuit
V  emf 1  et / RC 
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The RC Time Constant
Current in an RC circuit:
emf t / RC
I
e
R
When time t = RC, the current I drops by a factor of e.
RC is the ‘time constant’ of an RC circuit.
et / RC  e1 
1
 0.37
2.718
A rough measurement of how long it takes to reach final equilibrium
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