University of Puget Sound Introductory Physics Laboratory
6. Time-independent circuits
Name:____________________
Date:___________________
Objectives
1. To become familiar with analyzing circuits in terms of the node and loop
equations.
2. To experimentally investigate the rules of combination for resistors in series
and in parallel.
Equipment
Digital multimeter, bulbs, wires, batteries, holders, wires, resistance boxes, and
power supply.
Helpful Reading
Review the introduction to potential difference, current, and resistance in either
Hecht Chapter 17 or Rex/Jackson Chapter 21.
Background
To understand how electrical circuits work, it is helpful to learn two
simple rules about how current and voltage behave. It is also useful to keep the
more familiar gravitational analog in mind. In all of the circuits today the
current flow will be steady, that is, not changing in time. This occurs when the
elements in the circuit are purely resistive and can only dissipate but not store
energy. In the next experimnet we will study circuits involving elements that
can store energy, and hence the current flow is not necessarily steady.
The node equation. The total charge in a closed system is conserved. All charge
can do is flow around. Where a wire is joined to two or more wires, the current
in it may split (just like at a fork in a river). At such a junction, or node, the flow
inward must equal the flow outward. Mathematically stated, this is the node
equation: The total current leaving a node is equal to the total current entering the same
node. The node equation is also referred to as Kirchhoff’s first law. It is not a
separate law of physics but a restatement of the law of conservation of charge. It
is sometimes given in the alternate form: The algebraic sum of the currents at a node
is zero.
6-1
The loop equation. The total energy in a closed system is also conserved. The
potential energy of a charge just depends on where it is in a circuit. If it travels
around a loop, then the potential energy may go up and down, but if the charge
comes back to where it started it will have the same potential energy. This is just
like taking a bike ride. When you get back home you have the same
gravitational potential energy as when you started, no matter what hills or
valleys you encountered during you trip. In the context of circuits, this principle
is stated as the loop equation: The sum of the potential differences across all of the
elements along any closed path in a circuit is zero. The loop equation is also referred
to as Kirchhoff’s second law. However, it is just conservation of energy applied
to circuits. Sometimes it is given in the alternate form: The potential gains must
equal the potential drops around a closed loop.
In the circuit below, voltmeters (labeled V) are shown measuring the
potential difference across the A bulb, the C bulb, and the B-C combination. The
battery is labeled 3 V, for 3 volts.
V
B
C
V
3V
V
A
Notice that the voltmeters are now part of the circuit. In order that the
voltmeters not affect the circuit very much, what should be the resistance of the
voltmeters? Compared to what? Using the loop equation, determine the voltage
across bulb A. How many volts are across the B-C combination? How many
volts are across bulb C?
6-2
In the picture below, ammeters (labeled A1, A2, and A3) are measuring
the current leaving the power supply, the current through the A bulb, and the
current through the B-C bulbs.
A1
B
3V
A
C
A3
A2
Again, the measuring device is part of the circuit. In order that the ammeters not
affect the circuit significantly, what should be the resistance of the ammeters?
Compared to what? Think about the node equation. How should the currents
measured by the ammeters A1, A2, and A3 be related?
Exercises
In the following exercises, you will explore Kirchhoff’s laws using digital
multimeters to measure voltage and current in different time-independent
circuits. The DMM's (the yellow handheld instruments) should be familiar from
lab 3. Use a DMM for voltage measurements, in the DC volts mode. A voltmeter
will give opposite readings if you reverse the probes, so be aware of the signs of
the probes (+/-), and keep the relative sense of the probes the same as you go
around the circuit.
In the first exercises you will check out the loop equation in two different
circuits composed of batteries and bulbs. More complicate circuits can be built
up from these two basic variations. For these exercises you will not be
measuring the current. The resistance of the bulbs is comparable to the
resistance of the ammeter, thus the presence of the ammeter would have a large
effect on the circuit behavior.
6-3
Two bulbs in series. Connect two round bulbs and two batteries in series, as
shown below. Measure the potential differences across each element as you go
around the circuit. Start at a and measure Vab, then Vbc, Vcd, Vde, Vef, Vfg, and Vgh.
e
g
f
d
c
h
a
b
Vab
Vbc
Vcd
Vde
Vef
Vfg
Vgh
Vha
Sum
Do your results agree with the loop equation? What are the potential differences
across ab, cd, ef, and gh? Why? From now on you won't have to measure these.
How do the two bulbs share the voltage of the two batteries?
6-4
Two bulbs in parallel
Connect two round bulbs and two batteries in parallel, as shown below. Measure
the potential differences across each element as you go around the circuit. In this
circuit there are three loops. Measure all the voltage differences to fill in the table
below.
g
f
e
d
h
a
Vab
Vbc
Vce
Vef
Vfg
Vgh
Vha
Sum
0
c
b
Vab
Vbd
Vde
Vef
Vfg
Vgh
Vha
Sum
0
0
0
0
0
0
Vbd
Vde
Vec
Vcb
Sum
0
The three loops are easier to see (and to analyze) if we draw a schematic instead
of a picture of the circuit:
b
a
f
c
d
e
In the box below, write out all three loop equations symbolically, and then check
your equations with the data from your table.
6-5
Series circuits
Several resistors in series will behave as a single effective resistor.
......
R1
R2
R3
=
Reff
The effective resistance of a series combination is given by Reff  R1  R2  R3 .. .
Derive this result using the loop and node equations. Hint: Think about what
physical quantity is the same in a series.
Draw a schematic for a circuit that drives a current through a series combination
of two resistors. Where should you insert your measuring devices to measure
the current through and the voltage across each resistor? Draw them in, too.
6-6
Build the circuit you just designed, using the variable resistance boxes for your
two resistors in series. You should choose reasonably large resistors to avoid
drawing too much current from the power supply. For example, you might
choose 1 k and 5 k. While varying the power supply voltage, measure the
voltage across each resistor separately and across the series combination to
obtain three I-V curves. Take a sufficient number of data points to determine the
three resistances to within 1%. Graph your data on grid paper or using a
spreadsheet and insert it into your notebook. Determine the three resistances
and the uncertainty in the measured values and report them in the box below.
Does the series sum rule hold to within the accuracy of your measurements?
Current
voltage across R1
voltage across R2
voltage across pair
R1 = ___________________ Ω ± ___________________ Ω
R2 = ___________________ Ω ± ___________________ Ω
Reff = ___________________ Ω ± ___________________ Ω
What fraction of the power supply voltage drops across each of the resistors that
are in series? Is this fraction independent of the amount of current that flows?
Using the loop equation, derive this result in the general case. Series resistors
divide the voltage between them; this configuration is called a voltage divider.
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Parallel circuits
Several resistors in parallel will also act as a single effective resistor.
R1
R2
R3
......
=
Reff
The effective resistance of a parallel combination is given by
1
1
1
1



.. .
Reff R1 R2 R3
Derive this result using the loop and node equations. Hint: what quantity is the
same for parallel elements?
6-8
Draw a schematic for a circuit that drives a current through a parallel
combination of two resistors. Where should you insert measuring devices to
measure the current through and the voltage across each resistor? Draw them in
too.
Build the circuit you just designed, constructing a parallel combination from the
same resistors you used for the series measurement. While varying the power
supply voltage, measure the current through each resistor, the current leaving
the power supply, and the voltage across the parallel combination. Graph your
data as three I-V curves. Take a sufficient number of data points to determine
the three resistances to within 1%. Determine the three resistances and the
uncertainty in the measured values, and report them in the box below. Does the
parallel sum rule hold to within the accuracy of your measurements?
Current through
pair
Current through
R1
Current through
R2
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Voltage across
pair
R1 = ___________________ Ω ± ___________________ Ω
R2 = ___________________ Ω ± ___________________ Ω
Reff = ___________________Ω ± ___________________ Ω
What fraction of the total current from the power supply flows through each of
the resistors? Is this percentage independent of the supply voltage? Using the
loop and node equations, derive a general result for the current fraction for each
resistor in terms of the two resistances. Parallel resistors divide the current
between the two paths; this configuration is called a current divider.
Before you leave:
Show your instructor your data sets. Explain how you determined the
uncertainty in the measured values of the resistance.
Before the next lab:
Type a report on the work that you did in this lab. Your report should be about
one to two pages long (double spaced) with two attached plots for the series and
parallel I-V measurements. These can be drawn by hand on graph paper or
completed on the computer. Begin your report with an introductory paragraph
summarizing the conclusions in the body of the report. The body of the report
should describe the data that is used in the plots, discuss what the plots show,
6-10
and describe what can be concluded based on the experimental uncertainty. End
your report with a concluding paragraph. Be sure to proofread your writing for
proper grammar, punctuation, sentence structure, and general clarity.
6-11