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Current Electricity Unit Plan
Instructional Goals
1.
The electric field model developed previously is the agent for charge flow in a
conductor.
 A charge imbalance produces an electric field. The resulting potential difference can
be measured with a voltmeter.
 Without an external source of energy, charges move in such a way to restore the
balance of charge. (transient condition)
 With an external device (battery or generator) a charge imbalance can be sustained.
(steady state condition)
2.
Current is the "flow rate of charge" or quantity of charge per unit time passing a
cross-sectional area in the conductor.
 The ammeter is a device to measure flow rate.
 By contrast, drift velocity is the average speed of the individual charge carriers in the
conducting medium.
 Current is constant in a steady-state circuit.
3.
The internal structure of a material hinders the flow of charge driven by the
electric field. This is known as resistance.
 Despite the force applied by the electric field, charge quickly reaches a terminal
velocity in a steady-state circuit due to the interactions between the charge carriers
and the atoms in the conducting material.
 The amount of resistance is a function of the material (resistivity), the cross-sectional
area and the length of the conducting material; resistivity increases with temperature.
 In ohmic materials, resistance is the constant of proportionality between potential and
current
(V  I).
 In a non-ohmic material, potential is no longer directly proportional to current. (For
example: for a light bulb filament, V  I2).
4.
For simple series and parallel circuit arrangements, conservation of energy and
charge can be demonstrated.
 The energy dissipated by resistive elements in a circuit equals the energy provided by
the external source.
 The total quantity of charge moving in a circuit remains constant. The quantity of
charge in a given branch is inversely proportional to the resistance in that branch.
5.
The power dissipated by resistive elements is a function of both potential
difference and current.
6.
Representational tools include:


maps of surface charge distribution
schematic diagrams to represent circuits.
Sequence
1. Lab 1- What's happening in the wires?
2. Worksheet 1: Fields and potential difference in circuits
3. Lab 2: Charge distribution and potential difference
4. Resistance of Graphite Lines Lab
5. Lab 3: Discovering Ohm’s law
6. Lab 4: Series & Parallel Circuits
7. Water Analogy to Electric Circuits:
8. Quiz 2: Complete paths, light bulbs, ohm’s law
9. Worksheet 3 & 4: Series Circuit and Parallel Circuit Calculations
10. Extension: Series-Parallel Combination Circuits
11. Review
12. Test: Multiple Choice and Series-Parallel calculations part 2
OVERVIEW
Traditionally, circuits are treated as a separate case from electrostatics. When circuits are
introduced, the somewhat elusive concept of "potential difference" becomes even more
obscure and the meticulously developed concept of the electric field is abandoned
altogether. Students are left memorizing several "rules " (Kirchoff's laws, Ohm's law)
with little comprehension of the underlying physics. Vague thoughts of "voltage" as
some kind of "pressure" may be introduced, which not only confuses the cause of charge
motion with its effect, but is also utterly disconnected to prior treatments of electrical
potential.
The material developed here is an attempt to strongly bridge circuits with electrostatics.
Starting with the ideas (developed in Unit 2) that a separation of charge (the presence of
a net + or - charge) results in an electric field and in response to this field mobile charge
carriers move, students develop a more consistent model of circuits. In a circuit, the
battery (or other emf source) does work to maintain a charge separation, and thus creates
an electric field within the conducting elements ("conductors" or wires, and resistors).
At the resistors, mobile charges accumulate at one side, resulting in a charge separation
that produces a relatively large field compared to that within the conductor. Thus, charge
carriers experience most of their drop in potential at the resistors.
The rate of charge flow (current) is emphasized as a flow rate, or quantity of charge
moving through a cross-sectional area per unit time. This is distinguished from drift
velocity or average speed of a charge through a conductor. In a simple circuit, current is
the same in conductors and (series) resistors; drift velocity is not. In response to the
stronger electric field in resistors, drift velocity increases. Resistance in effect restricts
the area through which charge can flow. Thus, the restricted area through which charge
can flow effectively "evens out" the current.
Students develop a mathematical model for resistance by conducting a more-or-less
"typical" Ohm's Law Lab. They vary voltage supplied to a circuit containing an ohmic
(ceramic power resistor) and from the linear graph of V vs. I derive the mathematical
model for resistance. The conceptual model for resistance can be developed through an
analogy. Marbles rolling down an inclined pegboard studded with dowels or nails model
charge carriers traveling under the influence of a uniform field, encountering hindrance
from other material elements. The more hindrance, the lower the flow rate (for a given
field  tilt of the ramp).
Finally, students explore more complex circuits, containing two or more resistors in
series and/or parallel arrangement. Students are challenged to identify patterns relating
an overall quantity (e.g. voltage supplied) to resistor-specific quantities (e.g. voltage drop
across each resistor). With the ammeter and voltmeter, they collect data, and through
whiteboarding and discussion, the patterns are delineated. These patterns are then
checked against the models developed throughout to drive home the overall consistency.
Notes on circuitry conventions
Charge Flow Conventions:
The choice of direction of charge flow is left to the instructor. While we
recognize that electrons are the mobile charge carriers, the convention of discussing
current in terms of the flow of positive charge carriers (or holes) is favored by many
physics texts and is the preferred convention in electronics. The use of this convention is
due not only to the historical development but also to the fact we more naturally associate
the term "positive" with a surplus and "negative" with a deficit of charge. The way we
have defined the direction of the electric field and our designation of the positive terminal
of a battery as having greater electric potential favors the use of the Franklin convention.
Resistance in the wires of a circuit
During the introductory activities to this unit, it is not assumed that wires are ideal
conductors
(R = 0). The resistance (a term not used until the second lab) is found to be small, as seen
in small potential changes across the wires in the closed circuit of Lab 1. Later in the
unit, this small resistance is ignored in analyzing circuits. Each teacher will have to
decide how far to go in addressing the internal resistance of the connecting wires. This
can be circumvented, in the Ohm's Law Lab, by having the students measure the potential
across the resistor in the circuit, instead of across the battery.
1. Lab 1–What's happening in the wires?
This lab is divided into two sections; please refer to student lab sheets included in
materials.
Part I - Students monitor voltages in several parts of a circuit containing a charged
capacitor, and witness the transient field within the conductors during discharge.
Part II - Students discharge the capacitor through long and round bulbs. They notice
that the transient field takes significantly longer to disappear. The implication is
that the bulb “interferes” with the redistribution of charge carriers and the
restoration of the balanced distribution. This is visualized as the bulb filament
acting as a “bottleneck”, causing charge carriers to “pile up” as they pass through
the filament. These points should be drawn out of the discussion of student
models that should follow part II.
Purpose(s)
This introductory activity and the accompanying two-part lab is designed to lead students
from the concept of the electric field and electric potential to a more dynamic system
involving bulk motion of charge carriers within a conductor (current). By the end of this
lab, the students should be able to:
1. Recognize that charge carriers move in conductors as a result of electric fields set up by
an unbalanced distribution of charge within the conductors1 In the case of the discharging
capacitor, this imbalance is short lived, as charge is quickly redistributed.
2. Recognize a light bulb or other resistor constitutes a “bottleneck” that reduces the flow
of charge carriers, and “piles up” mobile charge on one side. As a result of the
unbalanced charge distribution across a resistor, a strong electric field exists in it.
3.
Recognize that the fields in the circuit (wires and resistors) do work on the charge
carriers, transferring their potential energy to other modes such as heat and light (and
charge carriers’ kinetic energy). Because the field in the resistors is so much stronger
than in the wires, almost all of the energy transfer (working) done in the circuit occurs in
the resistors. The case of the steady state circuit with a battery will be addressed in the
following lab.
Apparatus
- battery pack with three – D cells
- voltage probes with computer interface
- data collection software (Vernier Logger Pro, PASCO Science Workshop)
- capacitor (CASTLE 25,000 uF blue or 100,000 uF silver is ideal)
1
According to Sherwood and Chabay, the charge distribution responsible for the electric field is actually
on the surface of the conductor and forms a gradient along the length of the conductor proportional the
magnitude of the field. Additional information can be found in their textbook, Electric and Magnetic
Interactions, published by Wiley and Sons, 1995. A simplified treatment can be found in the optional
reading.
- mini light bulbs (#14 and #48)
Pre-lab discussion
NOTE: Conventional current — motion of positive charge—will be used throughout
this lab. If you prefer, you can cast all that is to follow in terms of electron flow.
The pre-lab discussion begins with the review of a capacitor made of two aluminum pie
pans separated by about 2 inches of air as first shown in unit 1.
Remind students that the uneven charge distribution produced an electric field which
exerted a force on the pith ball suspended between the plates. The excess charge on one
plate was transferred to the other by the action of the pith ball shuttling back and forth
between the plates. Now point out that we’ve only examined the field between the plates.
What would happen if the two plates were connected by a wire? Would a field exist in
the wire? Lacking the ability to place a test charge within wires can we deduce the
presence of a field within them? This lab should help students get a sense of what is
taking place in the wires.
Lab performance notes - Parts I and II
This lab can be performed as a teacher-led interactive demonstration, or with groups of
two or three students (preferable), depending on the availability of materials. It consists
of two parts, and a class-wide discussion is recommended, using whiteboards or other
similar devices, to pull the salient features out of each part. This may need to be done
after each part of the lab to be sure students are developing the concepts adequately.
Students will be using a voltmeter to detect the presence or absence of a field (stress the
concept developed at the end of Unit 2 that changes in electric potential only occur along
or against electric field lines). It is stated in the student materials, but worth
reiterating, that the existence (and relative strength) of an electric field will be
qualitatively inferred by the presence of a difference in potential. They can,
however, collect quantitative data on differences in potential, and indeed it is important
that they do so.
The ideal voltmeter for this activity is a set of voltage probes connected via an interface
(e.g. Vernier’s Lab-Pro or ULI or PASCO’s Science Workshop interface). This allows for
real-time collection of sometimes very transient (≈ 0.05 seconds) voltages, and allows
them be saved and later quantitatively examined. The voltages in this activity range
widely, from about 5 V to about 0.05V, easily dealt with by zooming in and out of a
voltage-time graph. For best results, sampling rates of 100-250 Hz have worked well, but
feel free to experiment.
Fig 1- Discharge of
capacitor through
connecting wires.
A digital voltmeter works too, but the transient voltages are difficult to read. The meter
will flash a value but neither the time for the voltage “spike” nor its “shape” can be
appreciated by the student. An analog meter will twitch, but that is even harder to read,
and the range of sensitivity required call for at least two meters (volts and millivolts).
It is imperative to place the black lead of the voltmeter or voltage probe at the starting or
reference point and to "lead with the red probe". In this way, increases in potential are
indicated by (+) voltmeter readings; (-) readings indicate that potential decreases.
Teachers are strongly encouraged to work through this lab prior to having the students
perform it in class.
Fig 2Change in potential
of connecting wire
when circuit is
completed.
Post-lab discussion
The lab parts are guided, but there should be some closure with the entire group. Groups
should prepare whiteboards of their final explanations of what is going on in the circuit
with the capacitor in it. In the ensuing discussion, it should be brought out that:
1. The battery does working on the charge carriers when" charging" the capacitor.
This moves charge from one plate to the other, creating an unbalanced
distribution until it is discharged.
2. The transient drop in potential from B to C needs to be accounted for. Note that
at C in the left diagram, the charge density is somewhat less than at B due to the
migration of charge across the boundary. This difference in charge density sets
up the field that drives the charge through the wires. When the charge is
distributed equally, the difference in potential between B and C returns to zero.
3. The "bottleneck" at the resistor results in a much, much stronger field (as
measured by a larger decrease in potential) than in the wires.
4. The field strength at different points within the circuit is a result of the
distribution of the charges at those points.
5. This would be a reasonable place to introduce the idea that surplus charge resides
on the surface of the conductor. Reading 1: charge distributions, which
describes the charge mapping, provides an excellent resource for students.
2. Worksheet 1: Fields and potential difference in
circuits
3. Lab 2–Charge distribution and potential difference
in circuits
Purpose(s)
This activity clarifies how a battery can maintain a constant charge flow (current) through
a simple circuit containing a resistor by working. It also develops charge distribution
mapping techniques as a tool for describing and predicting the fields within elements of
the circuit.
Apparatus
one battery pack with 3 D-cell batteries
6 Wires (2 of them with no insulation)
mini light bulbs (#14 and #48)
two bulb holders
digital voltmeter
Pre-lab discussion
Students should be reminded that in the previous lab
they learned that the charge distribution throughout the circuit is responsible for the field
that drives the current within the circuit.
This lab is an extension of lab 1, where the students used capacitors within a single
resistor circuit to describe the flow of charge due to separation of charges and the
resulting fields. In this activity students will use a battery pack to replace the charged
capacitor from lab 1.
Lab performance notes
Again, it is important to remind students that they should measure potential
differences “from black to red.” At this stage, it has not yet sunk in that (+) differences
represent gains and (-) differences represent losses in energy of the charge carriers.
Remind them that sign is as important as the reading of the magnitude on the multimeter.
Post-Lab discussion
After lab 1 and worksheet 1, the capacitor in the circuit is replaced by a battery.
Students will notice that the transient field observed in the conductor during the discharge
of the capacitor is replaced by a sustained field. The strong implication is that the
imbalance in charge carrier distribution is being actively maintained. The only “culprit”
possible for this is the battery. A quick “tour” of the circuit with the voltmeter reveals an
increase in electric potential across the battery (accompanied by a relatively strong
electric field), small decreases (and thus weak fields) in the conductors, and a large
decrease (and large field due to charge “pile
up”) at the bulb(s).
The use of different bulbs will result in
different drops in potential across the bulbs. At
right is a sample graph for a circuit in which
the charge moves first through a round (#48)
bulb before it moves through a long (#14) bulb.
Slight, but not negligible decreases in potential
should be observed in the wires as well.
Students may be perplexed by the fact that the
round (#48) bulb will failed to light (or barely
glow) while the long (#14) bulb is bright. The explanation lies in the much lower drop in
potential across the filament of the round bulb - the energy lost by the charge carriers is
insufficient to heat the filament to incandescence.
The final picture to be painted is that of the battery doing active work to maintain an
unbalanced charge distribution. This results in a continuous electric field that causes
mobile charge carriers to move in the conductors. The battery does work on the
charge carriers against an electric field, increasing the charges’ potential energy at the
expense of the battery’s chemical energy. One should make sure that students take
note of this. Batteries don’t run out of charge, but the ability to move the charge
carriers up-field. (If a Genecon or other hand-cranked generator is used, the charge
carriers’ potential energy is increased at the expense of the “cranker’s” mechanical
energy). Have the students add the potential differences moving around the circuit in
steps 6 and 7. If they made their measurements carefully, they should find that the
sum of the ∆V’s is nearly zero.
Because charge motion is present, it would be appropriate at this time to develop a model
of current. (This foundation needs to be laid before the activity developing Ohm's Law.)
One way to do this is to have students predict/describe the motion of charge carriers in the
wires as opposed to in the bulb filament. Their prediction should be along the lines of
"less motion" in the wires, owing to the weaker field present there. Ammeters can be
introduced as a "device to measure the motion of charge", and students can use them to
measure the current leaving the battery vs. the current leaving the bulb filament. Students
may be surprised to see it is the same.
Assure students their intuition is right on one count: the charge carriers must (on average)
be moving faster in the stronger field in the filament. Introduce the idea that the charge
carriers motion will be interfered with by the other components of the conductors (atoms,
other mobile charge carriers, etc.), but that in a stronger field, their average drift velocity
should be greater.
Point out that the wires are effectively much wider than the filament: the represent
a "wider hallway." Have students consider the quantity of charge carriers passing a
particular point in the wires during one second. They are moving slowly, but the hall is
wide. Now consider the quantity of charge carriers moving through the filament. They
are moving much faster, but their "hallway" is much narrower. Thus it is possible that in
both cases, the same number of charge carriers pass a point each second. Define this
q
"flow rate" as current: the quantity of charge passing a point each second I 
t . In a
resistor, the drift velocity is greater, but the current is the same as that in the wires. The
rate of charge flow (current) is emphasized as a flow rate, or quantity of charge moving
through a cross-sectional area per unit time. This is distinguished from drift velocity or
average speed of a charge through a conductor. In a simple circuit, current is the same in
conductors and (series) resistors; drift velocity is not. In response to the stronger electric
field in resistors, drift velocity increases. The increased velocity of the charge carriers
produces more collisions with the atoms in the conducting material, accounting for the
increase in temperature of conductors when charge passes through them. Resistance, in
effect, restricts the area through which charge can flow. Thus, the restricted area through
which charge can flow effectively "evens out" the current. The conceptual model for
resistance can be developed through an analogy. Marbles rolling down an inclined
pegboard studded with dowels or nails model charge carriers traveling under the
influence of a uniform field, encountering hindrance from atoms in the material. The
more hindrance, the lower the flow rate (for a given field).
This can be further demonstrated by a line of five students standing abreast (large
area for charge flow: a conductor) who move slowly to cross a line in one second, while
five students one behind the other (small area for charge flow: a resistor) must move
quickly. The flow rate is the same, while the drift velocity (people per second) differs
greatly.
4. Quiz 1
5. LAB 3 – Ohm's Law lab
Purpose
To investigate the relationship between voltage and current for a simple circuit with
constant resistance.
Apparatus
Variable voltage power supply or four D-cell and battery holder
Ceramic power resistors (5-50 ) or rheostat (different groups should have different
value resistors)
Connecting wires
Milliammeter and voltmeter or multi-meter or appropriate computer probes
One CASTLE “long” bulb (for optional extension)
diode (for optional extension)
Pre-lab discussion
Show students a simple circuit (bulb, battery and wire), and pose the question, “What do
you suppose is the relationship between the current and the potential difference that
causes the charge to move through the circuit?” Review the role of the battery: it does
work to maintain an imbalance of charge in the circuit. The greater the imbalance, the
stronger the field in the wires and the resistor, resulting in a stronger force on the mobile
charge carriers in the circuit. The stronger the force, the greater the flow rate.
Performance notes
If your students use the power supply, it is easy for them to adjust the voltage until the
ammeter reads a given value, then determine the potential drop across the resistor with a
multimeter. If they use a 4-cell battery pack, even though students will vary the potential
difference, making it the independent variable, you should encourage students to plot "V"
on the vertical axis. Indicate that if the graph is made this way it will provide more
useful information, not unlike the way we placed time on the horizontal axis in the ball
on an incline lab. Students can easily collect data on two different resistors, allowing
them to plot both sets of data on the same set of axes.
Optional extension - flashlight bulb
Have students perform the experiment again, this time with a long (#48) bulb instead of a
resistor.
Post-lab discussion
Most students readily see that the potential difference is proportional to the current and
that the slope is very nearly equal to the rated resistance of the resistor. Through a
discussion of the meaning of the slope, suggest that it represents the amount of energy
transferred to each coulomb of charge to result in a one ampere current. The greater the
value, the more the battery (or power supply) has to work for each ampere of current.
Thus, the slope is a measure of the circuit’s resistance to charge carrier motion.
Introduce “Ohm’s Law” as a re-arrangement of the V = RI equation they obtain from the
graph.. Use the term "ohmic" to describe a resistor with a linear dependence of current
on voltage.
If students collect data with the light bulb they will find that they will have to square I to
get a linear graph (V  I2). In the post lab discussion, challenge the students to account
for the difference in behavior between the bulb and the resistor. The thermal motion of
the filament’s atoms increases with the current, increasing the resistance to the movement
of charge carriers. Thus as the potential difference is increased, the increase in current
lags behind. Resistive elements that do not yield a linear plot are labeled "non-ohmic."
These data can also be used to develop the concept of electric power. Remind students
that in the past we've seen that the area under a graph can have physical significance.
Ask them to consider the meaning of the area under the V vs. I graph. Remind them that
the volts is a
joule/ coulomb, and the ampere is a coulomb/ second. An examination of the units
obtained when potential difference and current are multiplied reveals the units of power.
joule
coulomb joule


 watt
coulomb
s
s
The area under the Ohm's law curve is thus: P  1 2 Vfi nal  I . Note that this is average
power dissipated by the resistor as the voltage increased (since V  1 2 V f ). In circuits
where V doesn't change, P = VI. Manipulation of these two equations obtained from the
graph allows one to write the power equation in terms of I or V.
P  IV
P  I  IR substituting IR for V
P  I2 R
P  IV
V
V substituting
R
V2
P
R
P
V
for I
R
6. LAB 4 – Series and parallel arrangements
Purpose
After students have determined the relationship between potential difference, resistance
and current for a single resistor they examine these relationships with combinations of
resistors in series and parallel.
Apparatus
Variable voltage power supply or three D-cell and battery holder
Ceramic power resistors (5-50  - two per group)
Connecting wires
Milliammeter and voltmeter or multi-meter or appropriate computer probes
Pre-lab discussion
Sketch the circuit schematic above and ask what measurements could be made to
study how the combination of resistors might affect the current and potential difference at
different parts of the circuit. Students should be induced to measure the current before
R1, between R1 and R2, and after R2. They should measure the potential difference across
the battery, ∆VT , and across each resistor.
For the parallel circuit, they should measure the current in four places: before and after
the branches as well as in each branch.
Lab performance notes
Due to the number of rearrangements of the connecting wires that have to be made, it is
recommended that part of the class examine the series combination of resistors, while the
remainder examines the parallel combination. As an alternative procedure, the instructor
could assemble the circuit and make the measurements using current and voltage probes
connected to a computer with projection capability. The paired resistors should have
resistances that are simple integer multiples of one another (e.g., 5 and 15 , or 10 and
50 ).
Post-lab discussion
When students compare the data for the series circuit, they should be able to see charge is
conserved in its journey through the circuit, I1  I2  I3 , and that energy is conserved as
well VT  V1  V2  0 . Furthermore, the potential difference across each resistor is
V
R
proportional to the resistance: 2  2 . Since the magnitude of the gain in potential
V1 R1
across the battery is equal to the sum of the loss in potential across the two resistors, it
follows that the equivalent resistance of the circuit is simply the sum of the resistance
offered by the two resistors.
VT  V1  V2 
IT RT  I1 R1  I2 R2 
RT  R1  R2 
Substitute IR for V
Divide through by I since the current is the same everywhere
For the parallel circuit, students should also find that charge is conserved
Ibefore  I1  I2  Iafter , and that the current in each branch is inversely proportional to the
I
R
resistance 2  1 . Students will also see that the potential differences across each
I1 R2
branch are equal, but will usually find that they are smaller than the terminal voltage
provided by the battery. At first, they are troubled by this apparent non-conservation of
energy, but if they consider what happened in the first lab, they should realize that the
losses in potential in the connecting wires are not negligible. This is especially true if the
resistors used have relatively small resistance. The losses are more noticeable here than
in the series circuit since the current through the circuit is so much greater due to the
greatly reduced effective resistance.
In any event, below is a derivation for the expression for equivalent resistance for
resistors in parallel.
IT  I1  I 2
VT
V V
 1 2 
RT
R1 R2
1
RT

1
R1

1
R2

Substitute
V
for I
R
Divide out the V’s since VT  V1  V2 
7. Worksheet 2
This worksheet serves to deploy the relationship developed in labs 3 and 4.
8. Quiz 2
9. Worksheet 3
This worksheet gives students opportunities to apply the concepts to combination
circuits, as well as solve more complex AP-type problems.
10. Unit review
11.
Test