Verifying Ohm’s Law: A Portable and Innovative Method Using Two Smartphones

1. Introduction

In direct current (DC) circuits, as illustrated in Figure 1a, both the applied voltage and the current remain constant (Figure 1b). Ohm’s law, formulated by Georg Simon Ohm in 1827 [1], establishes a linear relationship between voltage and current in conductive materials that display ohmic behavior at a constant temperature. This relationship has been consistently validated in laboratory settings using commercial resistors of various values, provided that the temperature remains constant and the current does not exceed the manufacturer’s specified limits [2]. Mathematically, Ohm’s law is represented by the following formula:

$$∆V=RI,$$

(1)

where $∆V$, measured in volts (V), denotes the potential difference across the resistor; $R$ is the resistance measured in ohms (Ω); and $I$ is the electric current, measured in amperes (A), which indicates the magnitude of current flowing through the circuit. Digital multimeters and regulated voltage sources are commonly employed to demonstrate the relationships among these physical quantities and to verify Eq. (1).

In alternating current circuits, such as the example illustrated in Figure 1c, both voltage and current vary as functions of time (see Figure 1d). Ohm’s law defines the relationship between voltage (V), current (I), and impedance (Z), where V, I, and Z are represented as phasors.

Impedance quantifies the total opposition to alternating current, encompassing both the circuit’s resistance and the reactance of components such as capacitors and inductors. In circuits containing only resistive elements, the impedance (Z) equals the resistance (R), so Ohm’s law in the time domain maintains the same form as in direct current circuits:

$$v\left(t\right)=Ri\left(t\right).$$

(2)

Equation (2) describes a linear, instantaneous, and phase-shift-free relationship between voltage and current, as illustrated in Figure 1d. The current through the resistor (R) mirrors the time history of the voltage applied across its terminals, accounting for the absence of phase shift. When the voltage reaches its maximum, the current also reaches its maximum; similarly, when the voltage crosses zero, regardless of the slope’s direction, the current behaves identically.

In Figure 1c, if the voltage supplied by the source exhibits no initial phase shift, the potential difference across the resistor will similarly lack an initial phase shift. Consequently, given that $v\left(t\right)=V_{m}sen\left(wt\right)$, the current through the resistor, which represents the circuit current, can be expressed as:

$$i\left(t\right)={\displaystyle \frac{V_{m}sen\left(wt\right)}{R}}=I_{m}sen(wt).$$

(3)

Equation (3) shows that the instantaneous current has the same time dependence as the voltage, confirming the absence of a phase shift between them. Additionally, equation (3) shows that the maximum current in the resistor is directly proportional to the maximum applied voltage, which permits Ohm’s law to be formulated using the maximum values of these variables:

$$I_m={\displaystyle \frac{V_m}{R}}.$$

(4)

In an alternating current circuit containing several resistors in series, as depicted in Figure 2, the virtual oscilloscope displays the voltage and current (gray and yellow traces, respectively) across the reference resistor. This setup enables verification of Ohm’s Law using the relationship presented in equation (4). According to equation (4), the current through the reference resistor ($V_{m-ref}/R_{a-b}$) is 0.515 mA, which corresponds to the maximum current indicated on the virtual oscilloscope in Figure 2. The current can also be determined by dividing the maximum amplitude of the applied voltage by the total resistance of the circuit ($V_m/R_t$), thereby confirming the validity of Ohm’s Law when maximum instantaneous values of voltage and current are used in purely resistive AC circuits.

Analysis of the maximum voltage across the reference resistor in the circuit shown in Figure 2 demonstrates that increasing the number of resistive components decreases the maximum current. Consequently, the maximum voltage across the reference resistor also decreases proportionally. According to Ohm’s law (equation (4)) and given that the maximum current in the circuit equals that of the reference resistor, a relationship exists between the maximum voltage across the reference resistor and the total resistance of the circuit.

$$I_m={\displaystyle \frac{V_m}{R_t}}={\displaystyle \frac{V_{m-ref}}{R_{a-b}}}$$

(5)

The maximum reference voltage is as follows:

$$V_{m-ref}=\left(V_m{R}_{a-b}\right){{(R}_t)}^{-1}.$$

(6)

Equation (6) demonstrates that the maximum voltage across the reference resistor is inversely proportional to the total resistance of the AC circuit, assuming constant amplitude and frequency. This approach offers an alternative way to verify Ohm’s law in AC circuits without an AC current RMS meter by varying the circuit resistance and measuring the maximum voltage across a reference resistor.

In the past decade, mobile devices have become valuable tools for experimental work in physics and engineering. Their effectiveness stems from integrated sensors and specialized applications that support experiments in mechanics [3,4,5,6], optics [7,8,9,10], fluid mechanics [11,12,13], electricity and electronics [14,15,16], magnetism [17,18,19], and other fields.

This activity uses a purely resistive AC circuit to relate the voltage across a reference resistor to the total resistance of the circuit, thereby verifying Ohm’s law in a series resistor configuration. Following the methodology established by previous authors [15,16,20], the audio output of one smartphone serves as a signal generator, while the microphone input of another smartphone serves as an oscilloscope. The amplitude and frequency of the audio signal from the generator smartphone are kept constant while the circuit resistance is varied. The maximum amplitude detected across the reference resistor in each case is measured using the smartphone oscilloscope. Analysis of the results allows for a comparison between the experimental mathematical model relating to $V_{m-ref}$ and $R_t$ and the theoretical model predicted by equation (6).

2. Method

Figure 3 presents the circuit employed to demonstrate Ohm’s law. A Samsung Galaxy A04 (SM-A045M/DS) smartphone served as the signal generator, utilizing the “Tone Generator” function of the Physics Toolbox Suite app [21]. A Xiaomi Redmi Note 7 (M1901F7E) smartphone functioned as the oscilloscope via the “Audio Scope” feature of the Phyphox app [22]. Both devices ran on the Android operating system.

Each resistor was measured individually using a SparkFun 70C digital multimeter, with the measurement uncertainty estimated based on the instrument’s technical specifications [23]. A total of ten 0.47 kΩ resistors (nominal value) were required. To connect the mobile devices to the circuit, commercial 3.5 mm male TRRS connectors were utilized. Four-wire cables were soldered to these connectors, allocating one wire per channel: left audio, right audio, microphone, and ground. Finally, alligator clips were attached to the bare ends of the wires to ensure a reliable connection to the circuit.

Because smartphones are not specifically designed as laboratory measuring instruments, the output impedance of a smartphone functioning as a signal generator is nonzero, and the input impedance when used as an oscilloscope is finite. Consequently, both values needed to be measured.

The output impedance ($Z_{out}$) of the smartphone functioning as a generator is measured by applying a sinusoidal audio signal at a fixed frequency and constant amplitude. A second smartphone records the maximum no-load voltage (open-circuit voltage, $V_{m-oc}$) and the maximum load voltage ($V_{m-l}$) when a load resistor ($R_l$) is connected between the audio output and the signal generator’s ground. In this configuration, the load resistor and the output impedance of the audio generator form a voltage divider, allowing the application of equation (7) to determine the generator’s output impedance. Figure 4 presents the connection diagrams for the devices and the load resistor used in this procedure.

$$Z_{out}=\left({\displaystyle \frac{V_{m-oc}-V_{m-l}}{V_{m-l}}}\right)·R_l$$

(7)

A similar methodology was employed to determine the input impedance of the smartphone functioning as an oscilloscope ($Z_{in}$), with the modification that a known resistor was connected in series with the audio signal generator. In this configuration, the series resistance and the smartphone’s input impedance form a voltage divider, allowing the application of equation (8) to determine the unknown value.

$$Z_{in}=\left({\displaystyle \frac{V_{m-l}}{V_{m-oc}-V_{m-l}}}\right)·R_l$$

(8)

The output impedance of the smartphone functioning as a signal generator was measured across frequencies from 0.2 kHz to 1.0 kHz, using a load resistance of 47 Ω (measured value: 45.9 Ω). Similarly, the input impedance of the smartphone operating as an oscilloscope was evaluated over the same frequency range using a reference resistance of 4.7 kΩ (experimental value: 4.59 kΩ).

For all measurements, the amplitude of the signal generator was held constant by setting the mobile device volume to level 8, starting from zero intensity [20].

The circuit depicted in Figure 3 was assembled with a single resistor ($R_{ref}$), and the signal was measured on the smartphone oscilloscope between points a and b as indicated in the diagram. In this configuration, the total circuit impedance ($R_{a-b}{R}_{ref}$) is determined by the series combination of the smartphone signal generator’s output impedance ($Z_{out}$) and the parallel equivalent of $R_{ref}$ and the smartphone oscilloscope’s input impedance ($Z_{in}$).

$$R_{a-b}=Z_{out}+{\displaystyle \frac{R_{ref}·Z_{in}}{R_{ref}+Z_{in}}}$$

(9)

Resistors were sequentially added, modifying the circuit configuration as shown in Figure 3. The total resistance of the circuit was then calculated using equation (10):

$$R_t=R_{a-b}+R_{b-c}+R_{c-d}+\dots ,$$

(10)

where $R_{a-b}$ was obtained using equation (9), and the resistances $R_{b-c}$, $R_{c-d}$, and subsequent terms correspond to the measured values of each individual resistor.

For each configuration, the maximum voltage across the reference resistor was measured. Measurements were recorded each time the negative terminal of the audio generator was connected to positions c, d, e, f, and subsequent nodes indicated in Figure 3, while the audio output remained at position a. The data were exported in an Excel-compatible format for subsequent analysis, and screenshots were saved to document changes in the $V_{ref}$, signal as total resistance increased.

For each exported file, the voltage peaks of the sinusoidal signal were identified to calculate the average maximum voltage (V_m,ref${\overline{V}}_{m-ref}$) and its uncertainty. Subsequently, ${\overline{V}}_{m-ref}$ was plotted against $R_t$ to evaluate the empirical mathematical model against the theoretical model described in equation (6).

3. Results and Discussion

Figure 5 presents the output impedance ($Z_{out}$) of the smartphone operating as a signal generator and the input impedance ($Z_{in}$) of the device functioning as an oscilloscope. The experimental results demonstrate that both impedances remain constant and independent of the audio signal frequency within the analyzed range (0.20 kHz to 1.0 kHz). Furthermore, within this interval, both impedances exhibit purely resistive behavior, with no frequency-dependent variations attributable to reactive elements. This stability justifies using a single average impedance value across all calculations, thereby minimizing additional systematic errors.

Figure 6 displays screenshots obtained using the smartphone-based oscilloscope (a Xiaomi Redmi Note 7; see Figure 3) for three configurations of the total circuit resistance. In each case, the voltage signal is displayed as a function of time, as measured across the reference resistor. The signal frequency was maintained at 0.50 kHz for all measurements, with variations occurring only in amplitude.

Figure 6a, which represents the condition in which the total resistance is given by Equation (9), displays a sinusoidal signal with maximum amplitude. As the circuit’s total resistance increases, as illustrated in Figure 6b ($R_t=R_{a-b}+R_{b-c}$), 6c ($R_t=R_{a-b}+R_{b-c}+R_{c-d}$), and 6d ($R_t=R_{a-b}+R_{b-c}+R_{c-d}+R_{d-e}$) there is a clear and progressive decrease in the amplitude of $v_{ref}\left(t\right)$. This behavior directly demonstrates Ohm’s law in the time domain, indicating that the reduction in the $v_{ref}\left(t\right)$ amplitude results from the decrease in current $i\left(t\right)$ caused by the increased total resistance. The screenshots in Figure 6 further illustrate this trend: a greater number of resistors in series, and thus a higher $R_t$, produces a signal with a reduced amplitude, qualitatively confirming the behavior predicted by Ohm’s law.

The experimental results for ten total resistance configurations in the series circuit, constructed from identical resistors with a nominal value of 0.47 kΩ ± 5% (Figure 3), are summarized in Table 1. The total resistance for each configuration was calculated using equations (9) and (10) as more resistors were added. The total resistance uncertainty (${\sigma }_R$) arises from the propagation of uncertainties related to $Z_{out}$ and $Z_{in}$, combined with the measurement uncertainties of the individual components.

Signals were captured using the “Audio Scope” function of the Phyphox application. For each configuration, the average peak voltage (${\overline{V}}_{m.ref}$, in arbitrary units, a.u.) of the sinusoidal signal across the reference resistor was determined. The standard deviation (${\sigma }_V$) was used to quantify the dispersion of these amplitudes.

The results indicate a significant decrease in ${\overline{V}}_{m-ref}$ as $R_t$ increases, thereby qualitatively confirming the inverse relationship predicted by Ohm’s law. Figure 7 shows the average voltage across the reference resistor for each circuit total resistance configuration. The experimental data indicate a decreasing trend, confirming the inverse relationship between voltage and resistance, as suggested by the qualitative observations in Figure 6.

Fitting the data using a power function (orange line) yielded an expression of the form ${\overline{V}}_{m-ref}=A\cdot {R_t}^n$, with an exponent $n=-1.004$. This value approximates the theoretical exponent of −1, derived from Ohm’s law for a series circuit powered by a source of constant amplitude and frequency [Eq. (6)]. The relative discrepancy of 0.4% between the experimental and theoretical exponents is considerably smaller than the 10% limit commonly accepted in laboratory practices with uncalibrated instrumentation, demonstrating the precision achieved with the proposed methodology.

To verify the consistency of the proportionality constant $\mathit{A}$, the theoretical model from equation (6) was evaluated. As established in the methodology, considering the parallel and series impedance combinations with the external reference resistor ($R_{ref}$) in the circuit (Figure 3), the voltage measured between points a and b is ${\overline{V}}_{m-ref}=0.68683\mathrm{u}.\mathrm{a}$. Since the equivalent resistance $R_{a-b}$ is 0.396 kΩ is (Table 1), the expected theoretical product is 0.272 u. a.⋅kΩ. This calculated value closely matches the fitting coefficient $A=0.274u.a.·\mathrm{k}\mathsf{\Omega }$ derived from the power-law fit in Figure 7, exhibiting a relative difference of approximately 0.7%. This consistency, combined with the accurate experimental exponent ($n=-1.004$), provides robust validation of both the theoretical model and the experimental methodology.

The results indicate that Ohm’s law can be verified with high precision using only two smartphones as signal generators, an oscilloscope, commercial resistors, and TRRS cables. The internal consistency of the experiment, demonstrated by the agreement among the fitting exponent, the proportionality constant, and the directly measured values, underscores the reliability of mobile devices as experimental physics measurement tools, particularly in unconventional laboratory settings or remote learning environments.

4. Conclusion

The experimental results confirm the validity of Ohm’s law in series resistive circuits carrying alternating current, utilizing only two smartphones as a signal generator and an oscilloscope. The voltage amplitude measured across the reference resistor decreases in inverse proportion to the circuit’s total resistance, consistent with theoretical predictions. Fitting the mathematical model to the experimental data yields a fitting exponent of –1.004, which differs from the ideal value of –1 by only 0.4%. Similarly, the proportionality constant $\mathit{A}$ obtained from the power-law fit closely matches the value calculated directly from the measurements, differing by 0.7%. These minimal discrepancies, observed with uncalibrated mobile devices and standard commercial resistors (±5% tolerance), support both the validity of Ohm’s law and the accuracy and reliability of the smartphone-based methodology. This approach offers a portable, economical, and reproducible alternative to conventional laboratory equipment, making it suitable for resource-limited environments, distance education, and inquiry-based learning. The results underscore the potential of everyday mobile technology as a scientific tool for verifying fundamental principles of physics.