heart, when blood was leaving the lungs and entering the heart. Blood is a better conductor
than the tissues of the heart and lungs, so the motion of blood decreased the
Circuits
resistance
the heart and increased that of the lungs. This patient was healthy, but in
Circuits.
• Resistor of
We’ll look
at more
a patient
with
circulatory problems any deviation from normal blood flow would lead
complex
circuits
this
to week.
abnormal patterns
of resistance that would be revealed in such an image.
•
Capacitor Circuits.
A wire connected between the terminals of a battery carries a
and stretched, decreasing its cross-section area and
increasing its length. When the wire is reconnected to the battery, the new current is
AddingTO
a capacitor
to a
STOP
THINK 22.4
resistor circuit allows for
current.
The wire is removed
timing functions.
•
Electricity in the
Body. The physics of
A.
Larger than the original current.
resistor-capacitor circuits
B.
same
as the original current.
canThe
be used
to explain
signal
propagation
the original current.
C. Smaller thaninthe
nervous system.
22.5 Ohm’s Law and Resistor Circuits
23.2
729
Kirchhoff’s Laws
The relationship between the potential difference across a conductor and the current
passing through it that we saw in the preceding section was first deduced by Georg
Ohm’s
Law
23.4 Kirchhoff’s junction law.
an analyze
Our tools
and techniques
for FIGURE
Ohmit. and
is known
as Ohm’s
law:
hysical principles of potential differences
result of charge and current conservation,
the total current leaving the junction, as in
I
off’s junction law, which we wrote as
I3
Junction
∆V
=
I
R
Iout
Iin
1
(22.8)
I2
Iout
p. 108
p. 34
∆V
PROPORTIONAL
Junction law: I1 = I2 + I3
o
I
I
Ohm’s law for a conductor of resistance R
(23.1)
INVERSE
R
on law
w of nature. It’s an application of a law we
We can also apply the law of conservation
out gravitational potential energy in Chapial energy of an object depends only on its
that position. The same is true of electric
21 and as we discussed in ◀◀ SECTION 22.5 .
sed loop and returns to its starting point,
potential energy: ∆Uelec = 0. Because
ric
potential
around
any techniques
loop or closed
analyze
it. Our
tools and
for
FIGURE 23.5 Kirchhoff’s loop law.
(a)
23.2 Kirchhoff’s Laws
Vi = 0
R2
a
nature. It’s an application of a law we
pf law
e can also apply the law of conservation
gravitational potential energy in Chaprence of the ith component in the loop.
energy of an object depends only on its
if at least one of the potential differences
at position. The same is true of electric
need to explicitly identify which potential
1ative.
and as we discussed in ◀◀ SECTION 22.5 .
d loop and returns to its starting point,
otential energy: ∆Uelec = 0. Because
e
FIGURE 23.4 Kirchhoff’s junction law.
Graph of the potential
I3
Junction
around the circuit.
V
b
a
c
∆Vbattery
I1
Iin
∆V1
Iout
d
I2 2
∆V
e
Battery law:
Resistor
Junction
I1 = I2Resistor
+ I3
Distance along circuit
(23.1)
(23.2)
R1729
d
Kirchhoff’s
of a battery
and two resistors.Laws
If we start at
ative
terminal
of the
battery,
and plot the
sult of charge and
current
conservation,
raph
shown
in
the
figure.
The
potential
e total current leaving the junction,
as in
battery,
then
decreases
in
two
“downhill”
s junction law, which we wrote as
potential ends up where it started, as it must.
y to any circuit, as shown in FIGURE 23.5b. If
nd the loop formed by the circuit, the sum
law
c
b
sical principles of potential differences
hoff’s loop law:
Path around
the circuit
∆V2
(b) Add the potential
differences
around
the loop.
FIGURE 23.5 Kirchhoff’s loop law.
∆V1
(a)
Path around
the circuit
Start and
end here.
Loop
∆V3
c
b
∆V4
Loop law: ∆V1 + ∆V2 + ∆V3 + ∆V4 = 0
a
R1
d
R2
e
28/09/13 2:23 PM
What’s The Current?
The diagram below shows a segment of a circuit. What is
the current in the 200 Ω resistor?
What’s the Voltage?
The diagram below shows a circuit with two batteries
and three resistors. What is the potential difference
across the 200 Ω resistor?
Which is Brighter, Part I
a. Which bulb is brightest?
b. Which bulb is dimmest?
Suppose a wire is connected between points 1 and 2.
Does the brightness of each bulb:
! A. !Increase
! B. !Decrease
Which is Brighter, Part II
a. Which bulb is brightest?
Section 23.2 Kirchhoff’s
b. Which bulb is dimmest?
Laws
4. || In Figure P23.4, what is the current in the wire above the
junction? Does charge flow toward or away from the junction?
6V
I
1
Suppose a wire is connected between points 1 and 2.
2 Ωof each bulb:
Does the brightness
! A. !Increase
! B. !Decrease
5Ω
2.0 Ω
2
3.0 V
11
10 V
3
4
Key Principle
FIGURE P23.4
FIGURE P23.5
5. || The lightbulb in the circuit diagram of Figure P23.5 has a
Voltage
resistance of 1.0 Ω.
Consider the potential difference between
Is,
pairs of points in the
figure.
Current
a. What are the magnitudes
Flows. of ∆V12 , ∆V23 , and ∆V34 ?
b. What are the magnitudes if the bulb is removed?
6. | a. What are the magnitude and direction of the current in the
30 Ω resistor in Figure P23.6?
b. Draw a graph of the potential as a function of the distance
traveled through the circuit, traveling clockwise from
V = 0 V at the lower left corner. See Figure P23.9 for an
Another Complex Circuit
example of such a graph.
What is the current in the resistor?
30 Ω
9.0 V 6.0 V
18 Ω
3.0 V 6.0 V
12
13
14
15
FIGURE P23.6
FIGURE P23.7
7. || a. What are the magnitude and direction of the current in the
16
Reducing Complex Circuits, Part I
I
R2
R1
R3
Req = R1 + R2 + R3 + g
Rage Against the
Dying of the
Light.
Holiday lights use
simple series circuits
so that the bulbs can
be low voltage.
Suppose there are 50
bulbs in one string.
What is the voltage
across each bulb?
How does the string stay lit
when one bulb goes out?
Reducing Complex Circuits, Part II
∆V
R1
R2
R3
-1
1
1
1
+
+ gb
Req = a +
R1 R2 R3
Common Combinations
−1
Requivalent = R1 + R2 = 2R
Requivalent
⎡1
1 ⎤
R
=⎢ + ⎥ =
2
⎣ R1 R2 ⎦
What’s the Resistance?
There is a current of 1.0 A in the circuit below.
What is the resistance of the unknown circuit element?
What’s the Current?
What is the current supplied by the battery in the
following circuit?
Which is quicker?
Series
Parallel
Explain which wiring will result in quicker cooking, and why.
You must use a mathematical relationship.
A power and resistance puzzle
A 60 W and a100 W bulb are connected in series.
Think about the current in the circuit, and through each
bulb.
a. Which bulb has higher resistance?
b. When connected in series,which is brighter?
?
Circuits Calculations.
What is the current provided by the battery in the
following circuit?
Circuits Calculations.
What is the current through each of the resistors in the
following circuit?
Warming Up
What is the equivalent resistance of the following circuit?
Warming Up: What are the two resistors?
What are the resistances of the two unknown resistors
in this circuit?
R1
R2
Warming Up.
the figure.
Warming Up.
What’s the potential at the
the figure.
noted point in the circuit?
!
!
!
!
A. !
B. !!
C. !
D. !
10 V
6V
5V
4V
Warming Up.
What is the value of resistor R in the figure?
in
ci-
mF
mF
mF
in
circuit
is shown
in fiber
Figure
Is current
than,Ifless
I2 greater
myelinated
nerve
hasQ23.2.
a conduction
speed
of 55 m/s.
the
than,
or equal
to current
Explain.
I1 ?1.0
spacing
between
nodes is
mm and the resistance of segments
3. Current
flows
into
three
connected
together
one
betweenIinnodes
is 25
MΩ,
whatresistors
is the capacitance
of each
segment?
49.after
||| Athe
particular
axon has
nodes
spaced
0.80 mm
other asmyelinated
shown in Figure
Q23.3.
The
accompanying
apart.shows
The resistance
nodes isas20a function
MΩ; the of
capacitance
graph
the value between
of the potential
position.
insulated
segment
is 1.2
pF. What
the conduction
a.ofIseach
than,
less than,
or equal
to Iin ?isExplain.
Iout greater
nerve impulse
alongtothis
axon? the three resistances
b.speed
Rankofina order,
from largest
smallest,
50. | RTo
measure
signal
propagation
in
a nerve in the arm, the
1 , R2 , and R3 . Explain.
nerve is triggered near the armpit. The peak of the action potenR1
R2 then, R4.0
tial is measured at the elbow
and
ms later, 24 cm away
3
from the elbowIinat the wrist.
Iout
a. What is the speed of propagation along this nerve?
V of the speed made by measuring the time
b. A determination
between the application of a stimulus at the armpit and the
peak of an action potential at the elbow or the wrist would be
inaccurate. Explain the problem with this approach, and why
the noted technique is preferable.
Position
Q23.3
51.FIGURE
|| A myelinated
axon conducts
nerve impulses at a speed of
40 m/s. What is the signal speed if the thickness of the myelin
4. The
circuit
in Figure
has two are
resistors,
R1 7 R2 .
sheath
is halved
but noQ23.4
other changes
made towith
the axon?
Which resistor dissipates the larger amount of power? Explain.
R2
FIGURE Q23.4
a
ed
he
an
he
he
Ω
d?
I1
I3
a
c
I2
e
I5
I4
b
f
d
FIGURE Q23.8
9. a. In Figure Q23.9, what fraction of current I goes through the
3 Æ resistor?
b. If the 9 Æ resistor is replaced with a larger resistor, will the
fraction of current going through the 3 Æ resistor increase
decrease, or stay the same?
General Problems
52. || How much
R1 power is dissipated by
each resistor in Figure P23.52?
in
and I3 . Explain.
8. Figure Q23.8 shows two circuits. The two batteries are identica
and the four resistors all have exactly the same resistance.
a. Compare ¢Vab , ¢Vcd , and ¢Vef . Are they all the same? I
not, rank them in order from largest to smallest. Explain.
b. Rank in order, from largest to smallest, the five currents I1 to
I5 . Explain.
I
R1
R = 12 Ω
R21
9.0 V
R2 = 15 Ω
FIGURE P23.52
FIGURE Q23.5
53. |||| Two 75 W (120 V) lightbulbs are wired in series, then the
Howtohas
to
5. The
circuit in Figure
Q23.5
asolve?
battery
and two
with
combination
is connected
a 120
V supply.
Howresistors,
much power
Ris
Which
resistor
dissipates
the
larger
amount
of
power?
7
R
.
How
to
assess?
by each bulb?
1 dissipated
2
54.Explain.
|||| The corroded contacts in a lightbulb socket have 5.0 Ω total
6. Inresistance.
the circuit How
shown
in Figure
A and B are
much
actualQ23.6,
powerbulbs
is dissipated
by aglowing.
100 W
Then
the lightbulb
switch isscrewed
closed. What
happens
(120V)
into this
socket?to each bulb? Does it
stay the
same,
dimmer,
or go out? Explain.
55.get
|||| brighter,
A real battery
is not
just get
an emf.
We can
model a real 1.5 V battery as a 1.5 V emf in
series with a resistor known as the “inter1.0 Ω
nal resistance,” as shown in Figure P23.55.
A typical battery has 1.0 Ω internal resis1.5 V
tance due to imperfections that limit current through the battery. When there’s no
current through the battery, and thus no FIGURE P23.55
Measuring
Voltage and Current
voltage drop across the internal resistance,
the potential difference between
its terminals is 1.5
V, the value
Voltmeter:
Measures
Ammeter:
of the emf.
Suppose the terminals
of this battery
are connected to
potential
difference.
Measures
current.
a 2.0 Ω resistor.
R=0a.Ω;What
no voltage.
R=∞ Ω; no current.
is the potential difference between the terminals of the
battery?
b. What fraction of the battery’s power is dissipated by the
internal resistance?
a
FIGURE Q23.9
R
b
FIGURE Q23.10
10. Two of the three resistors in Figure Q23.10 are unknown bu
equal. Is the total resistance between
R
points a and b less than, greater than, or
equal to 50 Æ? Explain.
200 Ω
11. Two of the three resistors in Figure Q23.11
are unknown but equal. Is the total a
b
R
resistance between points a and b less
than, greater than, or equal to 200 Æ?
Explain.
FIGURE Q23.11
03/10/13 1:50 PM
What is the value of R?
50 Ω
9Ω
4th PROOF
What is the value of I?
What is the value of R?
R
3Ω
Voltage is. Current flows.
1)Which point,
a or b, is at a
higher
potential?
2.0 Ω
1.0 Ω
1.0 Ω
2.0 Ω
3.0 V
2)What is the potential difference Vab?
As Good As It Gets.
What is the current through each of the resistors in the
following circuit?
Capacitor formulas
Unit: farad, F:
Capacitors in series and in parallel
Which of the following combinations of capacitors has:
! 1) the highest capacitance?
! 2) the lowest capacitance?
Capacitor Discharge
τ is a “1/e life”
DVC
Half life:
t1 2 = τ ln 2
(DVC)0
e = 2.7
ln 2 = 0.69
0.37(DVC)0
0.13(DVC)0
0
0
t
2t
τ = RC
ΔVC = ( ΔVC )0 e−t /τ
3t
t
Intermittent Wipers
Rotating the dial changes a variable resistor. Does turning
the dial toward “F” increase or decrease the resistance?
RC Timing
The following circuits contain capacitors charged to 5.0 V. All of
the switches are closed at the same time.
After 1 second has passed, which capacitor has the highest
voltage? The lowest?
RC Timing
A 10 µF capacitor is initially charged to 20 µC. The
capacitor is discharged through a 1.0 kΩ resistor.
How long does it take to reduce the capacitor’s
charge to 10 µC?
Large Current, Short Pulse
A typical defibrillator has a 32 µF capacitor charged to
5000 V. The electrodes connected to the patient are
coated with a conducting gel that reduces the resistance
of the skin so that the effective resistance of the patient’s
torso is 100 Ω.
a. What is the current at the instant the switch is
closed?
b. What is the current 5.0 ms after the switch is
closed?
You Can’t, But the Monkey Can.
A reaction time challenge from Chapter 2.
Warming Up: Capacitance
1)What is the capacitance of the 2 l bottle Leyden jar
capacitors we use in class?
2)The capacitor is charged up to 10,000 V, then discharges
in 0.01 s. What is the current?
Area: ! ! !
Thickness: !!
κ:!! ! ! !
ε0:!! ! ! !
0.08 m2
0.00025 m
2.8
8.85x10-12 C2/N•m2
Warming Up: RC Timing
the figure.
A
B
Nerve Fibers (Axons) Can
Be Modeled As RC Circuits
Ion pumps and channels in the cell.
(A very simple model.)
K1 Exchange pump
Cell membrane
Conducting
fluids
Na1
K1
Na1
Sodium
channels
Potassium
channels
Resting potential of a nerve cell.
1
r
r
1
E50
r
E 1
2
2
2
1
r
r
E50
2
2
2
1
2
2
1
V (mV)
0
1
1
x
270
The electric
field inside the
cell membrane.
1
r
r
1
E50
r
E 1
2
2
2
A typical cell
membrane has a
thickness of 7.0 nm.
1
r
r
E50
2
2
V (mV)
0
1
2
2
1
What is the
strength of the
electric field inside
the cell membrane?
2
1
1
x
270
Action potential.
The action potential
Depolarization
Repolarization
2
2
1
2
1
1
1
1
1
1
Na
Reestablishing resting potential
1
2
Na1
1
2
1
2
1
1
1
2
2
3
1
140
t (ms)
0
270
2
2
DVmembrane (mV)
140
0
2
1
DVmembrane (mV)
140
1
2
2
1
DVmembrane (mV)
270
K1
2
2
2
1
2
K1
2
1
1
2
1
2
3
t (ms)
0
270
1
2
3
t (ms)
Signal propagation in the axon
Cell body
Nodes of Ranvier
Muscle fibers
Axon
Dendrites
Myelin sheath
Uninsulated axon
signal propagation
Sodium channels open
2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
DV
Cell body
x
Axon
Potassium channels open
1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1
2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2
2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2
1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1
DV
x
A wave of
potential travels
down the axon.
Membrane recovery
1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1
2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2
2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1
DV
v
x
Analyzing the axon, Part I: Resistance
What is the resistance along the length of a typical axon of length 1.0
mm and radius 5.0 µm? Ignore the resistance of the membrane, and
assume that the resistivity of the fluid inside the axon is 2.0 Ω•m.
Analyzing the axon, Part II: Capacitance
What is the capacitance of a typical axon of length 1.0 mm and radius
5.0 µm? The dielectric constant of the cell membrane is 9.0, and the
thickness is a typical 7.0 nm.
Analyzing the axon, Part III: Signal speed
For the typical axon of the previous two slides, how fast will an
action potential propagate from one end to the other?
Warmth Receptors:
Conduction Speeds
0.5 - 2.0 m/s.
Motor Neurons:
Conduction Speeds
80 - 120 m/s.
Analyzing the axon, Part IV: Increasing speed
If the radius of the axon is doubled, how will this affect the signal
speed?
R=
C=
ρL ρL
=
A πr2
κε 0 A κε 0 2π rL
=
d
d
RC ∝
1
r
The insulated axon: The myelin sheath
How does
insulating the axon
increase the signal
speed?
Explain this
in terms of the
time constant, the
resistance and the
capacitance of the
axon.
The
Insulated
Axon:
Saltatory
conduction
I
Nodes
Myelin sheath
Axon
I
I
I
Propagation
Nerve Fibers (Axons) Can
Be Modeled As RC Circuits
τ = RC = ( 25 MΩ ) (1.6 pF ) = 40 µs
Speed in insulated neurons
Depends on thickness of insulation; typical value:
Lnode 1.0 × 10 −3 m
v=
=
= 25 m/s
τ
40 × 10 -6 s
Similar for all mammals.
Energetics
Saltatory conduction is not only faster... it is also more
efficient; it uses less energy. Explain why.
(Think about the energy stored in the capacitance of the
cell membrane.)
The energy cost is not negligible; maintaining potentials in your
neurons requires 25%-40% of the energy use of your brain,
the most expensive organ in your body.
Kenneth C. Catania & Fiona E. Remple
Smaller animals are quicker on the uptake.
When a driver sees a red light, or a pedestrian stepping off the curb,
it takes about 0.65 seconds to hit the brakes. When a star-nosed
mole’s nose touches a potentially tasty treat, it takes a mere 0.23
seconds to determine if it is edible and, if so, to scarf it up.
Why are small animals blessed with such rapid reaction times?