The Electrostatic Potential Energy Formula E=qV in the Modular Unit Scaling Framework

J. Rogers, SE Ohio, 05 Mar 2025, 2357

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Abstract

This report explores the electrostatic potential energy formula $E=qV$ within the context of a modular unit scaling framework. By introducing a problem-specific scaling factor $V_{\text{Hz}}=\frac{q}{h}$, we demonstrate how voltage ($V$) can be scaled into frequency ($f$), mass ($m$), and energy ($E$) using a cascade of unit conversion factors. This approach not only simplifies the mathematics but also provides deep conceptual insight into the relationships between electrostatic potential energy, frequency, and mass. The framework reveals the fundamental unity of these quantities and highlights the flexibility and power of modular unit scaling in physics.  Because q is not fixed this means that the unit scaling of V to Hz varies as q changes. 

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1. Introduction

The electrostatic potential energy formula $E=qV$ is a cornerstone of electromagnetism, describing the energy associated with a charge $q$ in an electric potential $V$. Traditionally, this formula is treated as a standalone relationship, with little emphasis on its connection to other domains of physics, such as quantum mechanics and relativity. This report reframes $E=qV$ within a modular unit scaling framework, revealing its deep connections to frequency and mass through the introduction of a problem-specific scaling factor $V_{\text{Hz}}=\frac{q}{h}$.

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2. The Modular Unit Scaling Framework

The modular unit scaling framework expresses energy ($E$) in terms of unit conversion factors that bridge different physical domains:

• ${\text{kg}}_{\text{E}}=c^2$: Converts mass ($m$) into energy ($E$).

• ${\text{Hz}}_{\text{kg}}=\frac{h}{c^2}$: Converts frequency ($f$) into mass ($m$).

• ${\text{T}}_{\text{Hz}}=\frac{k}{h}$: Converts temperature ($T$) into frequency ($f$).

For electrostatic potential energy, we introduce a problem-specific scaling factor:

$${\mathrm{<span\; style="background-color:\; white;">V</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">q</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}$$

This factor converts voltage ($V$) into frequency ($f$) for a specific charge $q$.

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3. The Electrostatic Potential Energy Formula $E=qV$

The electrostatic potential energy formula $E=qV$ can be expressed in terms of modular unit conversion factors as:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">V</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">kg</span>}}_{\text{<span style="background-color: white;">E</span>}}$$

Here:

• $V_{\text{Hz}}=\frac{q}{h}$: Converts voltage ($V$) into frequency ($f$).

• ${\text{Hz}}_{\text{kg}}=\frac{h}{c^2}$: Converts frequency ($f$) into mass ($m$).

• ${\text{kg}}_{\text{E}}=c^2$: Converts mass ($m$) into energy ($E$).

This modular chain performs valid unit conversions, starting from voltage ($V$) and going through frequency, mass, and finally to energy.

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4. Calculating Frequency and Mass Equivalents

The modular structure of the framework allows you to calculate the frequency and mass equivalents of electrostatic potential energy without explicitly calculating $E$:

a. Frequency Equivalent:

$${\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">V</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">V</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">q</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">)</span>$$

• Units: Hertz (Hz).

• Physical Meaning: This represents the frequency scale that is equivalent to the electrostatic potential associated with voltage $V$ and charge $q$. It answers the question: If this electrostatic potential energy were converted into frequency (e.g., into a photon), what would the characteristic frequency be?

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b. Mass Equivalent:

$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">V</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">V</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">q</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">q</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

• Units: Kilograms (kg).

• Physical Meaning: This represents the mass scale that is equivalent to the electrostatic potential energy. It answers the question: If this electrostatic potential energy were converted into mass (e.g., through mass-energy equivalence), what would the equivalent mass be?

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c. Energy:

$${\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\mathrm{<span\; style="background-color:\; white;">V</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">V</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">kg</span>}}_{\text{<span style="background-color: white;">E</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">q</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">q</span>}$$

• Units: Joules (J).

• Physical Meaning: This recovers the standard formula $E=qV$ for electrostatic potential energy in Joules.

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5. Flexibility and Power of the Framework

This example demonstrates the flexibility and power of the modular unit scaling framework:

1. Handles Variable Properties: It can accommodate situations where scaling factors are not fixed constants but depend on variable properties of the system (like charge $q$).

2. Problem-Specific Adaptability: It allows you to define problem-specific scaling factors tailored to the particular context you are analyzing.

3. Maintains Modularity: Even with problem-specific factors, the modular structure of the unit conversion chain is maintained, allowing you to explore different domain equivalences systematically.

4. Conceptual Insight: It provides conceptual insight by allowing you to think about electrostatic potential energy not just in terms of energy itself, but also in terms of its equivalent frequency and mass scales.

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6. Why This Matters

The modular unit scaling framework has far-reaching implications:

• Demystifying Constants: By reinterpreting constants like $h$, $c$, and $k$ as unit conversion factors, we make their roles in physical laws much clearer and more intuitive.

• Unifying Physics: This framework highlights the deep unity of energy, frequency, mass, and other quantities, even when dealing with problem-specific scaling.

• Simplifying Conceptual Understanding: By reframing physical laws in terms of unit conversions, you make the physics more intuitive and accessible.

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7. Conclusion

The modular unit scaling framework provides a profound and elegant perspective on the electrostatic potential energy formula $E=qV$. By introducing the problem-specific scaling factor $V_{\text{Hz}}=\frac{q}{h}$, we reveal the deep unity of voltage, frequency, mass, and energy. This approach not only simplifies the mathematics but also provides deep conceptual insight into the relationships between these quantities.