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A Pedagogical MKS-based Electromagnetic Unit Convention with $\epsilon_0 = \mu_0 = 1/c$

This paper proposes a pedagogical, rationalized MKS-based electromagnetic unit system where vacuum constants are set to $\varepsilon_0 = \mu_0 = 1/c$ to simplify dimensional analysis and reduce cognitive load, a approach that preliminary student surveys suggest improves the clarity of electromagnetic derivations.

Original authors: Alisher Sanetullaev (New Uzbekistan University)

Published 2026-04-30

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CC BY 4.0

Original authors: Alisher Sanetullaev (New Uzbekistan University)

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 are trying to learn a new language, but every time you speak, you have to carry a heavy backpack filled with extra, unnecessary words. You know the grammar, you know the meaning, but the backpack makes every sentence clumsy and hard to remember.

This is exactly the problem a physicist named Alisher Sanetullaev sees in how we teach electricity and magnetism to college students.

The Problem: The "Backpack" of Constants

In standard physics classes, students use a system called SI units (the International System). To make the math work, this system requires students to carry around two specific "constants" (numbers) called $\epsilon_0$ and $\mu_0$ .

Think of these constants like two heavy, confusing backpacks.

They don't really change the story of the physics.
But students have to write them down, track them, and do math with them in almost every single equation.
This creates "cognitive load"—a fancy way of saying it tires out the brain's working memory. Students get so busy managing the backpacks that they forget to look at the actual scenery (the physics).

The Solution: The "Nu-System"

Sanetullaev proposes a new, simpler way to teach these concepts, which he calls the "Nu-system."

Instead of carrying two separate backpacks, he suggests we realize that these two constants are actually just different ways of writing the same thing: the speed of light ( $c$ ).

In his new system, he sets both constants to equal $1/c$ .

The Result: The heavy backpacks disappear.
The Benefit: The speed of light ( $c$ ) becomes the main character in the story, appearing clearly in every equation, rather than being hidden inside a messy math formula.

What Changes? (The Magic Tricks)

When you switch to this new system, some very strange and helpful things happen to the units we use:

Resistance Becomes "Weightless":
In the old system, electrical resistance (measured in Ohms) is a complex mix of mass, length, and time. In the Nu-system, resistance becomes dimensionless.
- Analogy: Imagine measuring the "heaviness" of a shadow. It has no weight, but it still exists. The author calls this the "VacOhm." It's like a "radian" for electricity—a pure number that tells you how much resistance a circuit has compared to empty space.
- Why it helps: Students can instantly see if a circuit is "stronger" or "weaker" than the vacuum of space without doing complex unit conversions.
Capacitors and Inductors Become "Time":
In the old system, a capacitor (which stores charge) and an inductor (which stores magnetic energy) have confusing units. In the Nu-system, they both simply measure time (seconds).
- Analogy: Think of a capacitor as a bucket that takes time to fill. In this system, the size of the bucket is the time it takes to fill. No complex math needed to figure out how long a circuit will take to react; you just look at the time value.
Radiation Pressure Gets Simple:
Calculating how much pressure light exerts on an object usually involves a messy fraction with many constants. In the Nu-system, it becomes as simple as multiplying the electric field by the magnetic field.
- Analogy: It's like going from a complex recipe that requires 10 specific measuring cups to just saying "mix the two ingredients."

Does It Actually Work?

The author didn't just dream this up; he tested it with real students at New Uzbekistan University.

The Test: He taught a group of students using the standard "backpack" method (SI) and then introduced them to the "Nu-system" for a few lectures.
The Result:
- Students felt significantly clearer about how electromagnetic waves (like light) work.
- They found the math easier to follow.
- Most students (over 50%) liked the idea of resistance being a simple number, though about 20% found the idea of "no units" a bit confusing at first.
- One student noted that seeing the speed of light ( $c$ ) directly in the equations helped them understand why light travels at that speed, rather than just memorizing a formula.

The Bottom Line

This paper isn't saying we should throw away the standard system used by engineers and scientists. The standard system is necessary for building bridges and designing circuits.

Instead, the author suggests that for learning physics, the "Nu-system" is a better pair of training wheels. It strips away the algebraic clutter so students can focus on the beautiful, simple logic of how electricity and magnetism actually work. It's like cleaning a foggy window so you can finally see the view clearly.

1. Problem Statement

The paper addresses persistent pedagogical difficulties undergraduate students face in electromagnetism, specifically regarding the cognitive load imposed by the International System of Units (SI).

The Core Issue: In standard SI, vacuum permittivity ( $\varepsilon_0$ ) and vacuum permeability ( $\mu_0$ ) are treated as two formally independent constants with distinct units. Students must simultaneously track the physical concepts and the "bookkeeping" of these constants during derivations.
Cognitive Load: According to Cognitive Load Theory (CLT), this dual tracking creates extraneous load, consuming working memory that should be dedicated to germane load (conceptual understanding).
Student Struggles: Research indicates students struggle to combine mathematical calculations with physical intuition, often failing to see the underlying unity of electromagnetic phenomena (e.g., why electromagnetic waves propagate at the speed of light $c$ ) because the relationship $c = 1/\sqrt{\varepsilon_0\mu_0}$ is obscured by algebraic complexity.

2. Methodology

The author proposes a new pedagogical unit convention ("nu-system") and validates it through a combination of theoretical derivation and preliminary empirical testing.

Theoretical Framework:
- Definition: The system sets vacuum constants to $\varepsilon_0 = \mu_0 = 1/c$ .
- Unit Conversion: A dimensionless conversion factor $\kappa = \sqrt{\mu_0 c} \approx 19.41$ is defined to map SI units to "nu-units" (e.g., $1 \text{ C} = \kappa \text{ nuC}$ , $1 \text{ A} = \kappa \text{ nuA}$ ).
- Scope: The system is designed as a theoretical companion for physics education, not a replacement for SI in engineering or experimental contexts.
Empirical Study:
- Participants: Two independent surveys administered to undergraduate physics students at New Uzbekistan University ( $N_1=46$ for baseline, $N_2=39$ for post-instruction).
- Procedure:
  - Survey A (Baseline): Measured confidence with SI constants, error frequency, and clarity of the SI wave equation.
  - Intervention: Students received a lecture and reading guide covering nu-system derivations for the wave equation, energy density, radiation pressure, and the Bohr atom.
  - Survey B (Post-Instruction): Measured perceived ease, clarity, and usefulness of the nu-system.
- Analysis: Statistical analysis used the Mann–Whitney U test (due to anonymous, unpaired data) and t-tests to compare means.

3. Key Contributions and Technical Innovations

A. Reformulation of Fundamental Laws

By substituting $\varepsilon_0 = \mu_0 = 1/c$ , the system simplifies Maxwell's equations and force laws while preserving the rationalized MKS structure:

Coulomb's Law: $F = \frac{c q_1 q_2}{4\pi r^2}$
Ampère's Law: $F/l = \frac{I_1 I_2}{2\pi r c}$
Maxwell's Equations: The displacement current term and source terms become cleaner. For example, Gauss's law becomes $\nabla \cdot \mathbf{E} = c\rho$ , and the Ampère-Maxwell law becomes $\nabla \times \mathbf{B} = \frac{\mathbf{J}}{c} + \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$ .

B. Dimensional Simplification

The most striking pedagogical feature is the reduction of electrical units to mechanical time units:

Resistance: Becomes dimensionless ( $[R] = 1$ $[R] = 1$ ). The impedance of free space is exactly $1$ "VacOhm" ( $v\Omega$ $v Ω$ ).
- Pedagogical Value: This allows students to immediately compare circuit impedance to vacuum impedance ( $Z_0 = 1 v\Omega$ ).
Capacitance and Inductance: Both acquire units of time (seconds).
- Pedagogical Value: Time constants ( $\tau = RC$ or $\tau = L/R$ ) become intuitively obvious as products of time and dimensionless numbers, eliminating the need to expand Farads and Henries into base SI dimensions ($kg, m, s, A$).
Charge and Current:
- $[nuC] = \sqrt{J \cdot s}$ (Square of charge has dimensions of action).
- $[nuA] = \sqrt{J/s} = \sqrt{Power}$ .

C. Simplified Derivations

The paper demonstrates "before/after" comparisons showing reduced algebraic steps:

Wave Equation: In SI, students must recall $v = 1/\sqrt{\varepsilon_0\mu_0}$ separately. In the nu-system, $\nabla^2 \mathbf{E} = \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2}$ makes $c$ immediately visible.
Radiation Pressure: Reduces from $p_r = \frac{|\mathbf{E} \times \mathbf{B}|}{\mu_0 c}$ to simply $p_r = |\mathbf{E} \times \mathbf{B}|$ .
Bohr Atom: The electron velocity derivation yields $v = \alpha c$ (where $\alpha$ is the fine-structure constant) immediately, without intermediate constant manipulation.

4. Results

Quantitative Findings

Wave Equation Clarity: Perceived clarity of the wave equation derivation improved significantly from a baseline mean of 2.91 (SI) to 3.51 (nu-system) on a 5-point scale ( $p = 0.005$ , Mann–Whitney U test).
Ease of Use: Students rated the nu-system derivations as significantly easier to follow than SI ( $\bar{x} = 3.46$ vs. midpoint 3, $p = 0.016$ ).
Preference: A majority of students (4.03/5) preferred the dimensionless resistance feature (VacOhm).
Error Reduction: Baseline data showed that 43% of students struggled with tracking $\varepsilon_0$ and $\mu_0$ , and 43% struggled with understanding what physical quantity these constants represented.

Qualitative Findings

Positive Feedback: Students reported that the system made Maxwell's equations "more symmetric" and helped them "see directly" why EM waves propagate at $c$ . The time-unit nature of $L$ and $C$ was cited as helpful for understanding LC oscillators.
Negative Feedback/Confusion:
- ~20% of students found the dimensionless resistance confusing or preferred familiar Ohms.
- Some students questioned the logical basis of setting $\varepsilon_0 = \mu_0$ , confusing a unit convention with a physical identity.
- Concerns were raised about portability to international contexts (exchange programs, conferences).

5. Significance and Conclusion

Pedagogical Impact: The paper provides preliminary empirical evidence that reducing extraneous cognitive load (by removing the tracking of two independent constants) improves student comprehension of core electromagnetic concepts.
Theoretical Transparency: The nu-system serves as a "theoretically transparent" framework that keeps the speed of light ( $c$ ) explicit at every stage of a derivation, bridging the gap between the algebraic simplicity of Gaussian/CGS units and the practical utility of MKS/SI units.
Limitations: The study is limited by its small sample size, single-institution setting, and unpaired survey design. The "transition cost" of learning a non-standard system is acknowledged as a real barrier.
Future Directions: The author suggests that this system should be introduced early in the course and used as a conceptual tool alongside SI, rather than as a replacement. Future research should employ controlled, paired designs to measure long-term retention and transferability.

In summary, the paper proposes a rationalized, MKS-based unit system that simplifies the algebraic structure of electromagnetism by unifying vacuum constants into the speed of light, demonstrating statistically significant improvements in student-perceived clarity and conceptual understanding.