# Lab 6: DC Circuits

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7 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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6 DC Circuits
Introduction
Kirchhoﬀ’s Rules
In Lab 5,you explored the behavior of circuits with resistors connected in series and in
parallel.You were able to analyze the behavior of these circuits using equations for equivalent
resistance.This approach does not generally work for circuits with more than one power
source.While equivalent resistance still has some application,in a multi-loop,multi-source
circuit like the one you will be investigating,you will need a more general approach to circuit
analysis —Kirchhoﬀ’s rules.
The ﬁrst rule,the loop rule,states that the total change in electric potential around a
closed loop must be zero.This means that the sum of the potential diﬀerences across all of
the devices in a closed loop must be zero,
￿
loop
ΔV
i
= 0 (1)
The second rule,known as the junction rule,applies to currents.This rule states that no
net current enters or leaves a junction,
￿
junction
i
i
= 0 (2)
These rules became evident in your measured potential diﬀerences and currents in Lab 5.
Here,you will use Kirchhoﬀ’s rules to predict the behavior of a more complex circuit.
Real Batteries:Internal Resistance
You will be investigating a multi-loop circuit powered by two D-cell batteries.In order to
make sense of the behavior of the batteries,you will need to account for the fact that real
batteries have internal resistance.This internal resistance reduces the potential diﬀerence
between the battery terminals whenever it is delivering current.

r
i
ΔV
Figure 1:Determining the internal resistance r of a battery.
If you measure the potential diﬀerence ΔV across the terminals of a battery as shown in
Fig.1 while it is delivering a current i,you will not measure its emf (E).Instead you will
measure
ΔV = E −ir (3)
where r is the internal resistance of the battery.When the battery is disconnected from the
circuit,so that i = 0,then a measurement of the potential diﬀerence across its terminals gives
E.It is therefore possible to calculate the internal resistance of a battery frommeasurements
of ΔV,i and E using
r =
E −ΔV
i
(4)
Experiment:a Multi-loop Circuit with Two Power Sources

2
R
2
R
3
i
3
i
1
R
1
i
2

1
r
1
r
2
B
CA
D
Figure 2:A multi-loop DC circuit with two power sources and three resistors.
You have been supplied with two D-cell batteries in holders,a set of three resistors mounted
on a Plexiglas stand,a digital multimeter (DMM),and several leads for connecting these
components.
1.Turn the knob on the DMMto measure resistance on the hundreds of Ω scale.Measure
and record the resistance of each resistor with the DMM by connecting one end of to
the V/Ω terminal and the other end to the COM terminal.Check your results with the
color codes.
2.Measure and record the emfs (E values) of your batteries.It is important not to connect
them to anything other than the DMM when you make these measurements.(Why?)
3.Construct the circuit shown in Figure 2.Make sure that the batteries’ orientations are
correct and that you record which E value goes with which battery.
4.Turn the knob on the DMM to measure potential diﬀerence on a scale of a few Volts.
Measure the potential diﬀerence ΔV
BA
= V
B
−V
A
between the points labeled A and B
in Fig.2 by connecting the V/Ω terminal of the DMMto point B and the COM terminal
of the DMM to point A.Similarly measure ΔV
CB
,ΔV
CA
,ΔV
DC
,and ΔV
DA
.
5.Use the DMM to measure magnitudes and directions the three unique currents in the
circuit i,i
1
,and i
2
.This will require you to connect the DMM in series with each
separate branch of the circuit.Remember never to connect an ammeter in parallel
with anything!Use the red current terminal on your DMMbest suited to measure mA
currents.Be careful not to change the way the circuit is arranged as you make these
measurements.If you’re not sure how,ask for help!
In order to determine the direction of the current,remember that a positive current
ﬂows into the red A (or mA) terminal and out of the black COM terminal.The directions
you measure may not agree with the arrows in Figure 2,and that’s OK!Record what
you actually observe.
6.Check your measurements by using your measured currents and voltages ΔV
CB
,ΔV
DC
,
and ΔV
CA
along with Ohm’s Law to calculate the resistances R
1
,R
2
,and R
3
.Com-
pare them with the actual values measured with the DMM in step 1.Resolve any
inconsistencies before you move on!
Analysis
1.Use Eq.4 and your measurements of ΔV
BA
and i
1
to ﬁnd the internal resistance r
1
of
battery 1.Similarly,use ΔV
DA
and i
2
to ﬁnd r
2
.

2
R
2
R
3
i
3
i
1
R
1
i
2

1
Figure 3:Simpliﬁed circuit diagram.
2.Use Kirchhoﬀ’s rules and your measured resistances and emfs to predict the magnitudes
and directions of each of the unique currents i
1
,i
2
,and i
3
.It simpliﬁes the mathematics
somewhat to combine the pairs of resistors connected in series R

1
= R
1
+ r
1
and
R

2
= R
2
+r
2
as illustrated in Fig.3.
Questions
1.Why was it important to measure the emfs (E values) of your batteries while they were
not connected to anything other than the DMM?
2.What are the internal resistances of your batteries?Show your calculations.
3.Are your measured potential diﬀerences compatible with the loop rule?
4.Show the calculations you made to predict the currents in the circuit shown in Figure 2.
Compare your predictions to your measurements,and discuss any signiﬁcant (> 5%)
discrepancies.