Validating the Ohm’s Law in Various Resistors Configurations
Tongwei Zhang
Case Western Reserve University
Abstract:
The aim of this experimental study was to investigate Ohm's Law and the behavior of
resistors in different configurations. Measurements of voltage and current were carried out using
a voltmeter, ammeters provided by the iOLab device, and various resistors. The main numerical
results showed minor disparities between theoretical and experimental current values, primarily
attributed to an unstable power source. Nevertheless, the experiment validated Ohm's Law and
highlighted the impact of series and parallel resistor arrangements on total resistance.
Introduction and background:
According to Ohm's law, the current through a circuit can be mathematically expressed as
follows:
𝑉 = 𝐼𝑅
Where V represents voltage, I refers to the current, R denotes the resistance.
For resistors, there exist two essential types: series resistors and parallel resistors. In the
case of series resistors, the total resistance 𝑅𝑒𝑞is calculated as:
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3
For parallel resistors, the equivalent resistance 𝑅𝑒𝑞 is obtained using the following expression:
1
1
1
1
= +
+
𝑅𝑒𝑞 𝑅1 𝑅2 𝑅3
To measure the voltage across a circuit, a voltmeter can be constructed using parallel
resistors with relatively high resistance. The use of high resistance resistors is crucial to ensure
that the current is not significantly affected during the voltage measurement.
On the other hand, to measure the current, an ammeter connect in series with low
resistance is utilized. In the context of the iOLab setup, although it inherently detects voltage, the
application of Ohm's Law in conjunction with resistance enables the determination of the current
flowing through the circuit.
Procedure:
The experiment involves the use of resistors with a specified error of 1%.
Experiment Part 1:
To determine the actual voltage of the iOLab device, a circuit is constructed with two 10k
ohms resistors connected in series, following the instructions outlined in the prelab manual. The
positive rail of the breadboard is connected to a 3.3V source with wire, and the negative rail is
linked to the ground, GND. Analog 7 is connected to the breadboard where both resistors are
inserted, resembling the configuration shown in Figure 1.
Figure 1 Voltage Divider Configuration
Next, the iOLab is activated to record the voltage value obtained from the voltage divider.
Figure 2 Voltage Value Obtained from the Voltage Divider
For the construction of an ammeter, two wires are attached to the high gain sensor
terminals G+ and G- of the iOLab device. One wire is linked to the 3.3V source, while the other
is connected to GND. Subsequently, a 10k ohm resistor and a one-ohm resistor are placed in
series. The purpose of such one-ohm resistor is to ensure the safety of the ammeter by preventing
excessive current from damaging the device. Figure 3 illustrates the setup for the ammeter
configuration.
Figure 3 Ammeter Setup with one 10k ohm resistor in series
Experiment part 2:
In this phase of the experiment, two 10k ohms resistors and one ammeter are arranged in
series, as depicted in Figure 4.
Figure 4 Configuration of Two 10k ohm Resistors with One Ammeter in Series
Subsequently, the voltage of the circuit is measured by placing the ammeter across the
ends of the one-ohm resistor, as illustrated in Figure 5.
Figure 5 Voltage Measurement Across Two 10k ohm Resistors Using the Ammeter
Experiment 3:
In this part, a series circuit is constructed using two 10k ohm resistors, one 4.7k ohm
resistor, and one 1-ohm resistor for ammeter measurement, as shown in Figure 6.
Figure 6 Configuration of Two 10k ohm Resistors, One 4.7k ohm Resistor, and One 1-ohm
Resistor in Series
To measure the voltage across this series circuit, the high gain sensor is employed, and
the voltage value is recorded and displayed, as illustrated in Figure 7.
Figure 7 Measured Voltage Value Across Two 10k ohm Resistors and One 4.7k ohm Resistor
Using the Ammeter
Accurate voltage readings are acquired using the high gain sensor, and the corresponding
values are documented for further analysis
Experiment 4:
In this experiment, a parallel circuit is established by placing two 10k ohm resistors in
parallel configuration, as depicted in Figure 8.
Figure 8 Configuration of Two Parallel 10k ohm Resistors
To determine the current flowing through the circuit, an ammeter is utilized. It is
observed that changing the positions of the ammeter does not affect the current measurement, as
shown in Figure 9.
Figure 9 Current Measurement Through Two Parallel 10k ohm Resistors Using the Ammeter
Experiment part 5:
In this phase of the experiment, the circuit is set up based on the circuit diagram provided
in the lab manual, as illustrated in Figure 10.
Figure 10 Circuit Diagram with Multiple Resistors
The experimental setup, as shown in Figure 11, is prepared accordingly.
Figure 11 Experiment Part 5 Setup
To measure the current flowing through points A1 to A3, the high gain sensor in conjunction
with a 1-ohm resistor is utilized. However, during the initial measurement with A1 positioned as
depicted in Figure 10, notable vibrations are observed when connected to the 3.3V source, as
shown in Figure 12.
Figure 12 Strong Vibrations Measured at A1
This phenomenon could potentially result from A1's proximity to the positive charge. In
response, A1 is relocated closer to the GND, which allows for the continued measurement of the
total current while yielding clearer data, as demonstrated in Figure 13.
Figure 13 Improved measured value for A1
For A2, the voltage measurement is captured and recorded as shown in Figure 14.
Figure 14 Measured voltage at A2
Similarly, the voltage measurement for A3 is documented as displayed in Figure 15.
Figure 15 Measured voltage at A3
Results and Analysis:
Experiment 1:
Based on the data acquired from the iOLab, the voltage measured across one 10k ohm
resistor is found to be 1.6509 ± 0.0036 V, as depicted in Figure 2. The uncertainty value of
0.0036 V is obtained from the standard deviation (σ) derived from the data. As two 10k ohm
resistors are connected in series, the voltage across them should be 3.3016 ± 0.0072 V.
Regarding the ammeter, a 1-ohm resistor is utilized as the resistance. Applying Ohm's
Law:
𝑉 = 𝐼𝑅
𝑉 = 𝐼 ∗ 1Ω
Consequently, the current measured by the ammeter has the same magnitude as the
voltage across the 1-ohm resistor.
The theoretical calculation of the current flowing through one 10k ohm resistor is
performed as follows:
𝐼=
𝐼=
𝑉
𝑅
3.3016𝑉
10000Ω
𝐼 = 0.33016 𝑚𝐴
Considering that all resistors possess an uncertainty of 1%, for a 10k Ω resistor δR = 100
Ω, the following uncertainties in current calculations are determined:
𝛿𝐼𝑣 =
|𝑉 +𝑅𝛿𝑉 ‒ 𝑅𝑉| = 0.00072 mA
𝛿𝐼𝑅 =
|𝑅 +𝑉𝛿𝑅 ‒ 𝑅𝑉| = 0.00327mA
2
2
𝛿𝐼 = 𝛿𝐼𝑣 + 𝛿𝐼𝑅 = 0.003 𝑚𝐴
Hence, the theoretical value of current is I = 0.330 ± 0.003 mA. The experimental value
obtained for the current is I = 0.302 ± 0.001 mA.
Experiment 2:
In this experiment, the resistors are arranged in series, resulting in a total resistance of R
= 10000 + 10000 = 20000 Ω.
𝛿𝑅1 = |𝑅1 + 𝛿𝑅1 + 𝑅2 ‒ 𝑅1 ‒ 𝑅2| = 𝛿𝑅1 = 100Ω
𝛿𝑅2 = |𝑅1 + 𝛿𝑅2 + 𝑅2 ‒ 𝑅1 ‒ 𝑅2| = 𝛿𝑅2 = 100Ω
𝛿𝑅 = 𝛿𝑅12 + 𝛿𝑅22 =
141 Ω
Thus, the total resistance is determined as R = 20000 ± 141 Ω.
With a measured voltage of 3.3016 ± 0.0072 V, the current would be:
𝐼=
𝑉
𝑅
3.3016𝑉
= 0.16508𝑚𝐴
20000Ω
𝐼=
The uncertainties in current calculation are as follows:
𝛿𝐼𝑣 =
|𝑉 +𝑅𝛿𝑉 ‒ 𝑅𝑉| = 0.00036𝑚𝐴
𝛿𝐼𝑅 =
|𝑅 +𝑉𝛿𝑅 ‒ 𝑅𝑉| = 0.00116mA
2
2
𝐼 = 𝛿𝐼𝑣 + 𝛿𝐼𝑅 = 0.00116𝑚𝐴
So, the theoretical value of current is I = 0.165 ± 0.001 mA. And the experimental value
obtained for the current is I = 0.145 ± 0.001 mA.
Experiment 3:
For Experiment 3, the resistors are connected in series, resulting in a total resistance of R
= 10000 + 10000 +4700 = 24700 Ω. Considering the resistor with a resistance of 4.7k ohms
introduces an uncertainty of 47 ohms.
𝛿𝑅1 = |𝑅1 + 𝛿𝑅1 + 𝑅2 + 𝑅3 ‒ 𝑅1 ‒ 𝑅2 ‒ 𝑅3| = 𝛿𝑅1 = 100Ω
𝛿𝑅2 = |𝑅1 + 𝛿𝑅2 + 𝑅2 + 𝑅3 ‒ 𝑅1 ‒ 𝑅2 ‒ 𝑅3| = 𝛿𝑅2 = 100Ω
𝛿𝑅3 = |𝑅1 + 𝛿𝑅3 + 𝑅2 + 𝑅3 ‒ 𝑅1 ‒ 𝑅2 ‒ 𝑅3| = 𝛿𝑅3 = 47Ω
𝛿𝑅 = 𝛿𝑅12 + 𝛿𝑅22 + 𝛿𝑅32 =
149Ω
Hence, the total resistance is determined as R = 24700 ± 149 Ω.
With a measured voltage of 3.3016 ± 0.0072 V, the current would be:
𝐼=
𝐼=
𝑉
𝑅
3.3016𝑉
= 0.13367𝑚𝐴
24700Ω
The uncertainties in current calculation are as follows:
𝛿𝐼𝑣 =
|𝑉 +𝑅𝛿𝑉 ‒ 𝑅𝑉| = 0.0002895𝑚𝐴
𝛿𝐼𝑅 =
|𝑅 +𝑉𝛿𝑅 ‒ 𝑅𝑉| = 0.0008035mA
2
2
𝐼 = 𝛿𝐼𝑣 + 𝛿𝐼𝑅 = 0.0008541𝑚𝐴
Therefore, the theoretical value of current is I = 0.134 ± 0.001 mA. And the experimental
value obtained for the current is I = 0.125 ± 0.001 mA.
Experiment 4:
In Experiment 4, the resistors are connected in parallel, resulting in an equivalent
resistance 𝑅𝑒𝑞 given by:
1
1
1
= +
𝑅𝑒𝑞 𝑅1 𝑅2
𝑅𝑒𝑞 = 1
𝑅1
𝛿𝑅1 =
𝛿𝑅2 =
|
|
1
1
+ 𝑅
1
1
1
+ 𝑅
𝑅1 + 𝛿𝑅1
2
1
1
1
+ 𝑅 + 𝛿𝑅
𝑅1
2
1
2
= 5000 Ω
‒1
𝑅1
‒1
𝑅1
1
1
+ 𝑅
2
1
1
+ 𝑅
2
| = 24.87 Ω
| = 24.87 Ω
𝛿𝑅𝑒𝑞 = 𝛿𝑅12 + 𝛿𝑅22 = 35.17Ω
So, 𝑅𝑒𝑞 = 5000 ± 35 Ω.
With a measured voltage of 3.3016 ± 0.0072 V, the current would be:
𝐼=
𝑉
𝑅
3.3016𝑉
= 0.66032𝑚𝐴
5000Ω
𝐼=
|
𝑉 + 𝛿𝑉 𝑉
‒ = 0.00144𝑚𝐴
𝑅
𝑅
𝛿𝐼𝑅 =
|𝑅 +𝑉𝛿𝑅 ‒ 𝑅𝑉| = 0.00459mA
𝛿𝐼𝑣 =
|
2
2
𝐼 = 𝛿𝐼𝑣 + 𝛿𝐼𝑅 = 0.004811𝑚𝐴
Hence, the theoretical value of current is I = 0.660 ± 0.005 mA. The experimental value
obtained for the current is I = 0.605 ± 0.001 mA.
Experiment 5:
A1:
According to the circuit diagram provided in the lab manual, resistor R1 is in
parallel with R2 and R3, while the entire system of R1, R2, and R3 is in series with R4.
The total resistance 𝑅𝑡 can be calculated as follows:
𝑅𝑡 = 1
1
1
𝛿𝑅𝑡1 =
𝛿𝑅𝑡2 =
|(
1
1
+ 𝑅 +𝑅
𝑅1 + 𝛿𝑅1
2
3
|(
1
1
+ 𝑅 + 𝑅 + 𝛿𝑅
𝑅1
2
3
2
1
1
+ 𝑅4
1
+ 𝑅 +𝑅
𝑅
2
3
)
= 10651.42 Ω
+ 𝑅4 ‒ ( 1
)
𝑅1
+ 𝑅4 ‒ ( 1
𝑅1
1
1
+ 𝑅 +𝑅
2
1
1
+ 𝑅 +𝑅
2
+ 𝑅4 )
|
= 35.52 Ω
+ 𝑅4 )
|
= 16.57 Ω
3
3
|(
𝛿𝑅𝑡3 =
𝛿𝑅𝑡4 =
|(
1
1
1
+ 𝑅 + 𝑅 + 𝛿𝑅
𝑅1
2
3
3
1
1
1
+ 𝑅 +𝑅
𝑅1
2
3
)
+ 𝑅4 ‒ ( 1
𝑅1
)
1
+ 𝑅4 )
1
+ 𝑅 +𝑅
2
+ 𝑅4 + 𝛿𝑅4 ‒ ( 1
𝑅1
3
1
2
= 7.936 Ω
+ 𝑅4 )
1
+ 𝑅 +𝑅
|
3
𝛿𝑅𝑡 = 𝛿𝑅𝑡12 + 𝛿𝑅𝑡22 + 𝛿𝑅𝑡32 + 𝛿𝑅𝑡42 =
|
= 47.25 Ω
61.90 Ω
Therefore, 𝑅𝑡 = 10651 ± 62 Ω.
With a measured voltage of 3.3016 ± 0.0072 V, the current would be:
𝐼=
𝐼=
𝛿𝐼𝑣 =
𝑉
𝑅
3.3016𝑉
= 0.3100𝑚𝐴
10651Ω
|𝑉 +𝑅𝛿𝑉 ‒ 𝑅𝑉| = 0.0006760𝑚𝐴
𝛿𝐼𝑅 =
|𝑅 +𝑉𝛿𝑅 ‒ 𝑅𝑉| = 0.001791mA
2
2
𝐼 = 𝛿𝐼𝑣 + 𝛿𝐼𝑅 = 0.001914𝑚𝐴
Thus, theorical value of I is 0.31 ± 0.02𝑚𝐴. And the experimental value is
𝐼=
0.287 ± 0.0002𝑚𝐴.
A2:
The current of a parallel circuit is inversely proportional to the resistance.
Therefore, the current 𝐼2 at A2 can be calculated as follows:
𝐼2 = 𝐼 ×
𝑅2 + 𝑅3
= 0.1840𝑚𝐴
𝑅2 + 𝑅3 + 𝑅1
The experimental value of current at A2 is 0.178 mA.
A3:
The total current is conserved in the circuit. So,
𝐼3 = 𝐼 ‒ 𝐼2 = 0.126𝑚𝐴
The experimental value is 0.118mA.
Error Analysis:
Ensuring a constant voltage supply of 3.300V ± 0.0036 throughout the experiment proves
challenging due to potential fluctuations when the battery is turned on and off. Furthermore, as
the experiment progresses from one to five, the battery depletion may lead to a decrease in the
supplied voltage. To mitigate these voltage variations and enhance experimental reliability,
employing a wire with a constant current source is recommended. Utilizing a constant current
source would stabilize the electrical supply, minimizing the impact of voltage fluctuations and
resulting in more consistent and accurate experimental measurements.
Discussion:
By comparison, it is evident that for all five experiments, the experimental and theoretical
values of current slightly differ. Despite the minor discrepancies observed, overall, the data
aligns with Ohm's Law, supporting its validity. The uncertainties in resistors and voltage
measurements contribute to the variations observed between theoretical and experimental values.
As expected, using resistors in series increases the total resistance, but decreases in parallel. This
behavior is consistent with the principles of series and parallel resistors.
Conclusion:
To conclude, the experiment aimed to investigate Ohm's Law and the behavior of
resistors in various configurations. Although theoretical and experimental current values have
minor discrepancies, primarily attributed to an unstable power source, the experiment
successfully confirmed the validity of Ohm's Law and demonstrated the expected effects of
series and parallel resistor arrangements on total resistance. For future experiments, it is
advisable to use a stable power supply, such as a constant current source, to enhance accuracy
and reliability.
Acknowledgements:
Reference:
Driscoll, D., General Physics II: Electricity and Magnetism, “Pre lab: DC Circuits”
Driscoll, D., General Physics II: Electricity and Magnetism, “DC Circuits”