iClicker Quiz
(1) I have completed at least 50% of the reading and
study-guide assignments associated with the lecture,
as indicated on the course schedule.
A. True
B. False
Hint: this is a good time to read the pitfalls in the margins of
the text. Current is NEVER used up.
Today: Review of resistance use video demos and MU.
Also  POWER  &
A. New Circuit elements: Batteries . B. Terms: EMF,
terminal voltage, internal resistance, load resistance.
C. How to add up resistors in series and parallel: light bulb
problems.
B. How many of these terms are in this problem? EMF,
terminal voltage, internal resistance, load resistance.
C. Do we know enough to do part a? pp. A true, B. No
D. If so what is the equation which helps us?
Circuit Analysis
•Simple electric circuits may contain batteries,
resistors, and capacitors in various combinations.
•For some circuits, analysis may consist of
Today’s work
combining resistors.
•In more complex complicated circuits, Kirchhoff’s
Rules may be used for analysis.
– These Rules are based on conservation of energy and
conservation of electric charge for isolated systems.
•Circuits may involve direct current or alternating
current.
Introduction
Direct Current
•When the current in a circuit has a constant
direction, the current is called direct current.
–Most of the circuits analyzed will be assumed to
be in steady state, with constant magnitude and
direction.
•Because the potential difference between the
terminals of a battery is constant, the battery
produces direct current.
•The battery is known as a source of emf.
Section 28.1
EMF “Electromotive Force”
•The electromotive force (emf), ε, of a battery
is the maximum possible voltage that the
battery can provide between its terminals.
–The emf supplies energy, it does not apply a
force.
•.
Section 28.1
Batteries as circuit elements
• A battery or other constant-voltage device (e.g. power supply)
is usually the energy source in a direct-current (DC) circuit.
• The positive terminal of the battery has higher potential than
the negative terminal.
• The electromotive force (emf, ε) of a battery is the opencircuit voltage between its terminals when no current is
flowing.
• Ideally, the battery has no internal resistance of its own.
• We often idealize the wires in a circuit to have zero resistance.
+
−
Real batteries
1. Chemical energy: an electrolyte solution allows negative
ions to flow toward and react with the anode (−), while
positive ions flow toward and react with the cathode (+).
2. Charge build-up prevents the reaction from proceeding
unless an external circuit allows electrons accumulating
at the anode to return to the cathode.
3. Dead when the reactants are used up. Rechargeable if the
anode/cathode reactions are reversible.
4. Internal resistance tends to increase with age, use
and multiple recharge cycles.
Typical Alkaline (Zn/MnO2)
1.5 to 1.6 V open circuit, 1.1 to 1.3 V closed circuit.
163 W-hr/kg (590 J/g) – 400 kJ total for AA
0.034 (122.4 C) to 15 A-hr (54 kC) of charge depending on size.
85% Capacity after 4 years of non-use.
Lithium Ion
3.2 V open circuit, 2.5 to 3.0 V closed circuit.
230 W-hr/kg (828 J/g) – 460 kJ total for AA
0.160 (576 C) to 1.4 Amp-hrs (5.04 kC) of charge
95% capacity after 5 years of non-use
NiCd
1.2 V open circuit
50 W-hr/kg (180 J/g) – 140 kJ total for AA
70% capacity after one month of non use (500-5000 cycles)
Zn-Air bus battery:
200 W-hr/kg (720 J/g)
320 kW-hr (1.15 GJ) of energy
Internal Battery Resistance, r
•If the internal resistance is zero, the
terminal voltage equals the emf.
•In a real battery, there is internal
resistance, r.
•The terminal voltage, ΔV = e – Ir
•The emf is equivalent to the open-circuit
voltage.
– This is the terminal voltage when no
current is in the circuit.
– This is the voltage labeled on the battery.
•The actual potential difference
between the terminals of the battery
depends on the current in the circuit.
Section 28.1
Load Resistance
•The terminal voltage also equals the voltage
across the external resistance.
–This external resistor is called the load
resistance.
–In the previous circuit, the load resistance is just
the external resistor.
–In general, the load resistance could be any
electrical device.
• These resistances represent loads on the battery since
it supplies the energy to operate the device containing
the resistance.
Section 28.1
Internal resistance in non-ideal batteries
Terminal voltage:
V ≡ Vab = ε − I r < ε
Current:
ε
V ε
= <
I=
R+r R R
Two AA batteries yield a combined ε = 3 V. You observe a
terminal voltage of 2.7 V while delivering 300 mA of current to
an ultra-bright flashlight.
What is the load resistance (R) ? (1) 1Ω (2) 3Ω (3) 9Ω (4) 10 Ω
What is the internal resistance of a single AA battery?
(1) 0.5Ω (2) 1Ω (3) 2Ω (4) 3Ω
Power: where does it go?
•The total power output of the battery is
 P = I ΔV = I ε
•This power is delivered to the external resistor
(I 2 R) and to the internal resistor (I2 r).
•
P = I2 R + I2 r
•The battery is a supply of constant emf.
–The battery does not supply a constant current
since the current in the circuit depends on the
resistance connected to the battery.
–The battery does not supply a constant terminal
voltage.
Section 28.1
Listening quiz
A. A battery is a source of constant EMF
B. A battery is a source of constant terminal
voltage
C. A battery is a source of constant current
to the load.
Consider this problem: Do we know enough to do it?
How do you approach it?
Are any of the resistors in Series with others?
Are any in parallel?
Resistors in Series
•When two or more resistors are connected end-toend, they are said to be in series.
•For a series combination of resistors, the currents
are the same in all the resistors because the
amount of charge that passes through one resistor
must also pass through the other resistors in the
same time interval.
•The potential difference will divide among the
resistors such that the sum of the potential
differences across the resistors is equal to the total
potential difference across the combination.
Section 28.2
Resistors in Series, cont
•Currents are the same
– I = I1 = I2
•Potentials add
– ΔV = V1 + V2 = IR1 + IR2
= I (R1+R2)
– Consequence of
Conservation of Energy
•The equivalent resistance
has the same effect on the
circuit as the original
combination of resistors.
Section 28.2
Equivalent Resistance – Series
•Req = R1 + R2 + R3 + …
•The equivalent resistance of a series
combination of resistors is the algebraic
sum of the individual resistances and is
always greater than any individual
resistance.
•If one device in the series circuit creates
an open circuit, all devices are inoperative.
Section 28.2
Equivalent Resistance
– Series – An Example
•Are all 3 representations equivalent?
A. yes
B. no
Two resistors are replaced with their equivalent
resistance.
Section 28.2
Some Circuit Notes
•A local change in one part of a circuit may
result in a global change throughout the circuit.
–For example, changing one resistor will affect the
currents and voltages in all the other resistors and
the terminal voltage of the battery.
•In a series circuit, there is one path for the
current to take.
•In a parallel circuit, there are multiple paths
for the current to take.
Section 28.2
Resistors in Parallel
•The potential difference across each resistor
is the same because each is connected
directly across the battery terminals.
ΔV = ΔV1 = ΔV2
•A junction is a point where the current can
split.
•The current, I, that enters junction must be
equal to the total current leaving that junction.
– I = I 1 + I 2 = (ΔV1 / R1) + (ΔV2 / R2)
–The currents are generally not the same.
Section 28.2
–Consequence of conservation
of electric charge
Equivalent Resistance –
Parallel, Examples
•Are all three diagrams equivalent?
•Equivalent resistance replaces the two
original resistances.
Section 28.2
– Parallel Resistors
•Equivalent Resistance
1
1
1
1
=
+
+
+
Req R1 R2 R3
•The inverse of the
equivalent resistance of two
or more resistors connected
in parallel is the algebraic
sum of the inverses of the
individual resistance.
– The equivalent is always less
than the smallest resistor in
the group.
Section 28.2
Resistors in Parallel: Final Observations
•In parallel, each device operates independently of
the others so that if one is switched off, the others
remain on.
•In parallel, all of the devices operate on the same
voltage.
•The current takes all the paths.
– The lower resistance will have higher currents.
– Even very high resistances will have some currents.
•Household circuits are wired so that electrical
devices are connected in parallel.
Section 28.2
The switch is initially open. When the switch is closed,
the current measured by the ammeter will:
A. increase B. decrease C. stay the same D. fall to zero.
Combinations of Resistors
•The 8.0-W and 4.0-W
resistors are in series and can
be replaced with their
equivalent, 12.0 W
•The 6.0-W and 3.0-W
resistors are in parallel and
can be replaced with their
equivalent, 2.0 W
•These equivalent resistances
are in series and can be
replaced with their equivalent
resistance, •14.0
Section 28.2
W
Consider this problem: Now do we know enough to do it?
How do you approach it?
Are any of the resistors in series with others?
Are any in parallel?
Which part involves doing an integral?
A. a
B. b
C. both a and b.
D. neither. E. Squirrel
Resistance of an object with arbitrary shape
End-to-end:
dz
ρ
 length 
dR = d  ρ
dz
=
=ρ
2
2
A(z )
π (b − a )
 area 
R = ∫ dR =
Inside-out:
ρ
L
π (b 2 − a 2 ) ∫0
dz =
ρL
π (b 2 − a 2 )
dr
dr
 length 
=
ρ
=
ρ
dR = d  ρ

2π r L
A(r )
 area 
ρ b
ρ
b
ln
R = ∫ dR =
dr
=
 
2π L ∫a
2π L  a 
Resistance of an object with arbitrary shape
dz
a
b
L
R = ∫ dR = ∫ ρ
= ρ∫
L
0
=
dL
dz
dz
= ρ∫
= ρ∫ 2
π r ( z)
A
A( z )
ρ  L  L du
dz
= 
 ∫z =0 2
2
π [a + (b − a ) z / L] π  b − a  u
ρ  L  1 1  ρ L
 −  =

π  b − a  a b  π ab
Effective resistance: two resistors in series
R1
R2
⇒
R1 and R2 experience the same
current but different voltages.
Largest R has largest V.
V V1 + V2 I R1 + I R2
=
= R1 + R2
Req = =
I
I
I
Req is larger than
either R1 or R2.
Effective resistance: two resistors in parallel
R1
⇒
R2
R1 and R2 experience the same
voltage but different currents.
Smallest R has largest I.
Req =
1
1 
V
V
V
=
=
=  + 
I I1 + I 2 V / R1 + V / R2  R1 R2 
−1
Req is smaller
than smallest
of R1 and R2.
Reduction of a resistive network
R
R
R
R
R
R
⇒
R
R
R
R
⇒
2R
R/2
R
2R
⇒
Reduction of a resistive network
R
R
b
R
R
b
a
a
⇒
R
R
a
R
b
R
R
⇒
R
R/3
⇒
(7/3)R
R
R
Reduction of a resistive network
Apply 42 V between a and c.
What is I between a and c?
I=3A
What is Vbc?
Vbc = 6 V
What is I2?
I2 = 2 A
Compare the brightness of
the four identical bulbs in
this circuit.
R
R
R
V
D is in parallel with a zero-resistance wire. The current will
take the zero-resistance path and bypass D altogether.
A and B are in series. So they will burn equally bright.
Together, they see the full battery voltage.
C experiences the full battery voltage, or twice the voltage
experienced by A or B. So C is four times as bright.
If R1 is removed, R2 will glow
(1) more brightly.
(2) less brightly.
(3) same brightness as before.
If R1 is removed, R2 will glow
(1) more brightly.
(2) less brightly.
(3) same brightness as before.
Household devices are wired to run in parallel!
Strings of 50 Christmas lights in
series. Assume ~100 V source
and 25 W power consumption.
http://www.ciphersbyritter.com/RADELECT/LITES/XMSLITES.HTM
What is the resistance of a single bulb?
A. 2Ω
B. 4Ω C. 8Ω
D. 10 Ω
Gustav
Kirchhoff
•1824 – 1887
•German physicist
•Worked with Robert
Bunsen
•Kirchhoff and Bunsen
– Invented the spectroscope
and founded the science
of spectroscopy
– Discovered the elements
cesium and rubidium
– Invented astronomical
spectroscopy
Section 28.3
Multi-loop circuits
ε
Branch: An independent current path experiences only one current
at a given moment. It may be a simple wire or may also contain one
or more circuit elements connected in series.
Junction: A point where three or more circuit branches meet.
Loop: A current path that begins and ends at the same circuit point,
traversing one or more circuit branches, but without ever passing the
same point twice.
Multi-loop circuits
ε
The circuit above has: 3 branches, 2 junctions, 3 loops.
To solve for 3 unknown branch currents, we need 3 equations.
To get these equations, use all but one (2 − 1 = 1) junction, and as
many independent loops as needed ( 3 − 1 = 2).
Kirchoff’s current rule: ∑ I n = 0
Current rule: The total current flowing into a junction is zero.
Arrows define positive branch-current directions. A current later
determined to flow opposite its arrow is “negative”.
ε
+ I − I1 − I 2 = 0
+ I − I1 + I 2 = 0
+ I + I1 + I 2 = 0
Kirchoff’s Voltage Rule: ∑ ∆Vn = 0
Voltage rule: The voltage changes around a loop sum to zero.
Arrows define positive branch-current directions.
∆V = +ε for a battery crossing from – to + terminal.
Use ∆V = –I R when crossing a resistor in the positive direction.
Use ∆V = +I R when crossing a resistor in the negative direction.
Alpine loop elevation
+ ε − I1 R1 = 0
ε
Single-loop circuit example
+ 20 − I (2000) − 30 − I (1000)
− I (1500) + 25 − I (500)
= 15 − I (5000) = 0
I
I = 15 / 5000 = 3 mA
VA − Vground = +20 − I (2000) − 30 − I (1000)
= −10 − I (3000) = −10 − (0.003)(3000)
= −19 V
Multiloop circuit example
Bottom loop:
I2 = −I1
+ 5 − I 2 (1) − I 3 (2) = 0
Substitute I2 in junction Eq:
I3 = −2I1
+ I 2 − I1 − I 3 = 0
Substitute I2 and I3 in top loop:
I1 = −5/3
Solve for currents:
I1 = −5/3, I2 = 5/3, I3 = 10/3
+ 5 + I1 (1) + I 2 (1) − 5 = 0
Multiloop circuit example
+ 6 I1 − 10 − 4 I 2 − 14 = 0
I 2 = 1.5 I1 − 6
− 2 I 3 + 10 − 6 I1 = 0
I 3 = −3I1 + 5
+ I1 + I 2 − I 3 = 0
I1 + (1.5 I1 − 6) − (−3I1 + 5)
= 5.5 I1 − 11 = 0
I1 = 2 I 2 = −3 I 3 = −1
EMF “Electromotive Force”
•The electromotive force (emf), ε, of a battery
is the maximum possible voltage that the
battery can provide between its terminals.
–The emf supplies energy, it does not apply a
force.
•The battery will normally be the source of
energy in the circuit.
•The positive terminal of the battery is at a
higher potential than the negative terminal.
•A battery is like a current pump. + charges
are pumped from - terminal to +.
•We consider the wires to have no resistance.
Section 28.1