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Estimation of nanometer-thickness layer of 6B graphite: an experimental activity to study Ohm's law in higher education

This paper presents an experimental activity for higher education that validates Ohm's law by using voltage and current measurements to estimate the nanometer-scale thickness of 6B graphite traces, yielding results that closely align with scanning probe microscope data.

Original authors: Paulo Henrique Eleuterio Falsetti, André Coelho da Silva, Leonardo Geraldino da Silva, Douglas Mendes da Silva Del Duque, Murilo Antonio Menegati, Bruno Fernando Gianelli, Vagner Romito de Mendonça, I

Published 2026-06-04

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CC BY 4.0

Original authors: Paulo Henrique Eleuterio Falsetti, André Coelho da Silva, Leonardo Geraldino da Silva, Douglas Mendes da Silva Del Duque, Murilo Antonio Menegati, Bruno Fernando Gianelli, Vagner Romito de Mendonça, Idelma Aparecida Alves Terra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a piece of tracing paper and a standard 6B pencil. You draw a thick, dark line across the paper. To the naked eye, that line looks like a flat, two-dimensional smear. But in reality, it's a tiny, three-dimensional mountain range of graphite dust, only a few hundred nanometers high (a nanometer is one-billionth of a meter).

This paper describes a clever classroom experiment that turns a simple pencil drawing into a physics lesson about Ohm's Law (the rule that governs how electricity flows through wires) to measure the height of that invisible graphite mountain.

Here is the story of how they did it, broken down into simple steps:

1. The Problem: The "Invisible" Height

The researchers wanted to know exactly how thick the layer of graphite was. Usually, to measure something that thin, you need a super-expensive, high-tech microscope called a Scanning Probe Microscope (SPM). Think of the SPM as a "blind man's cane" for the microscopic world; it physically touches the surface to feel the bumps and measure the height.

But the researchers wanted to see if they could figure out the same height using only a pencil, a piece of paper, a battery, and some basic math.

2. The Analogy: The Graphite as a "Squished Wire"

To understand their method, imagine electricity flowing through a copper wire. The wire has a length, a width, and a thickness. The thicker the wire, the easier it is for electricity to flow.

When you draw a line with a pencil, you are essentially creating a very short, very flat "wire" made of graphite.

Length: How long the line is.
Width: How wide the line is.
Thickness (Height): How tall the pile of graphite is.

The researchers knew the Length and Width of their line. They didn't know the Thickness. However, they knew that the "resistance" (how hard it is for electricity to pass through) depends on all three of these dimensions.

3. The Experiment: The "Pencil Circuit"

Here is what they actually did:

Step A: Calibrating the Pencil. First, they took a whole pencil lead (the part inside the wood), measured its diameter, and ran electricity through it to figure out exactly how "resistive" that specific brand of 6B graphite is. This gave them a baseline number, like knowing the "density" of the material.
Step B: Drawing the Line. They drew a very specific line on tracing paper, pressing hard to lay down 50 layers of graphite.
Step C: The Electrical Test. They connected the ends of their drawn line to a battery and a meter. They changed the voltage (the "push" of the electricity) and measured how much current (the "flow") went through the line.
Step D: The Math Magic. Because they knew the relationship between voltage, current, and resistance (Ohm's Law), and they knew the length and width of the line, they could use a simple formula to solve for the missing piece: the thickness.

It's like knowing how fast a car is going and how long the trip took, and then calculating the distance without ever measuring the road.

4. The Results: Two Methods, One Answer

The team did this calculation and got a result: The graphite layer was about 456 nanometers thick.

To check if they were right, they took the same drawing to a partner institution and ran it through the fancy SPM microscope (the "blind man's cane"). The microscope measured the height directly and said: "It's about 440 nanometers thick."

5. The Conclusion

The two numbers (456 and 440) are very close. When you account for the tiny errors in measurement (like the slight wobble of a hand drawing a line or the precision of the ruler), the results overlap.

What does this mean?
The paper claims that you don't always need a million-dollar microscope to measure nanometer-scale things. By using a simple pencil, some paper, and the fundamental laws of electricity, students can accurately estimate the thickness of a graphite layer.

The authors suggest this is a great activity for university physics classes because it connects abstract math (Ohm's Law) to a tangible, real-world object. They also note that with some simplification, it could even work for high school students.

In short: They used electricity to "weigh" a pencil drawing and found that the math matches the high-tech microscope. It's a low-cost, high-smart way to teach physics.

Technical Summary: Estimation of Nanometer-Thickness Layer of 6B Graphite

Problem Statement
Ohm's Law is a fundamental concept in electrical engineering and physics; however, literature indicates that students at various educational levels struggle to grasp its abstract concepts, particularly the relationships between voltage, current, and resistance. This difficulty often stems from the lack of tangible connection between theoretical formulas and physical reality. While previous studies have utilized pencil graphite traces for educational experiments, many lack rigorous statistical treatment suitable for higher education or fail to validate their results against precise metrological standards. Furthermore, the roughness of standard paper often complicates thickness measurements, making it difficult to determine the nanometer-scale dimensions of deposited graphite layers accurately.

Methodology
The authors propose an experimental activity designed for Higher Education (with potential adaptation for High School) that utilizes Ohm's Law to estimate the thickness of 6B graphite traces deposited on tracing paper. The methodology involves a multi-step process:

Resistivity Determination: The electrical resistivity ( $\rho$ ) of the specific 6B graphite core (Faber-Castell) was first determined. This involved measuring the diameter and length of the graphite rod, applying a known voltage, and measuring the resulting current to calculate the slope of the $I$ vs. $V$ relationship.
Graphite Deposition: A 6B pencil was used to draw a continuous, homogeneous trace on tracing paper (Canson A4, 60 g/m²). The deposition consisted of 50 layers drawn in a single direction to ensure consistency.
Geometric Measurements: The width ( $b$ ) of the graphite trace was measured at various intervals (0–2 cm to 8–10 cm) using a smartphone camera and ImageJ software, calibrated against a digital caliper.
Thickness Calculation via Ohm's Law: The trace was treated as a resistor with a rectangular cross-section ( $A = b \times h$ , where $h$ is the unknown thickness). By measuring current and voltage across different lengths ( $L$ ) of the trace and utilizing the previously determined resistivity, the authors applied the linear relationship derived from Ohm's Law ( $I = \frac{b \cdot h}{\rho \cdot L} V$ ) to solve for $h$ .
Validation via SPM: To validate the electrical method, the thickness of the same graphite traces was measured using a Scanning Probe Microscope (SPM) (Shimadzu SPM-9700HT). Topographical images were processed using Gwyddion software to extract altimetric profiles and calculate the step height between the paper surface and the graphite layer.

Key Results

Resistivity: The resistivity of the 6B graphite was calculated as $1.857(19) \times 10^{-4} \, \Omega\cdot\text{m}$ . The authors note this value is higher than that of pure graphite due to the clay and binder content in the pencil lead.
Electrical Thickness Estimation: Using the proposed methodology, the average thickness of the graphite layer was determined to be 456.5(34) nm.
SPM Validation: The SPM measurements yielded an average thickness of 440(50) nm.
Statistical Agreement: The authors performed a statistical comparison of the two methods. The difference between the means (16.5 nm) was found to be statistically insignificant, as the uncertainty intervals of both methods overlap (423.1–489.9 nm for the electrical method vs. 390–490 nm for SPM). The difference of $16(50)$ nm includes zero, confirming the compatibility of the two measurement techniques within experimental uncertainties.

Significance and Claims
The paper claims that the proposed methodology is a viable, accessible, and effective alternative for estimating the thickness of nanometer-scale conductive layers without requiring expensive equipment for the primary measurement. By validating the electrical method against high-precision SPM data, the study demonstrates that standard laboratory equipment (multimeters, power supplies, and basic imaging) can yield results consistent with advanced microscopy when proper statistical analysis is applied.

The authors position this work as a significant pedagogical tool for Higher Education Electricity courses. It offers a concrete application of Ohm's Law that moves beyond abstract circuit analysis to the physical characterization of materials. The study emphasizes that the activity encourages active student participation, connects theory to practice, and fosters significant learning by allowing students to derive physical properties (thickness) from fundamental electrical measurements. The authors suggest that with appropriate adaptations, this approach could also be implemented in High School curricula to demystify electrical concepts and introduce nanoscale measurement techniques.

1. The Problem: The "Invisible" Height

2. The Analogy: The Graphite as a "Squished Wire"

3. The Experiment: The "Pencil Circuit"

4. The Results: Two Methods, One Answer

5. The Conclusion

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