The Astonishing Connection: Converting Coulombs to Hertz via 1 / h

 J. Rogers, SE Ohio, 10 Mar 2025, 2115

Right now I think this is the result of them defining 1V = 1J/C.  This results in being normalized to both c^2 and h in a way that makes this the result.  But that means that both E/h = f and C/h = f, that means that C is already the energy value for 1V.  

In the world of physics, fundamental constants like Planck’s constant ($h$), the speed of light ($c$), and Boltzmann’s constant ($k$) often seem like arbitrary numbers—mere conversion factors between our chosen units. However, when we peel back the layers of the SI system and express these constants in terms of natural unit conversions, an astounding picture emerges. One striking example is the conversion of the SI unit of electric charge (the coulomb, $C$) into natural frequency units (Hertz, $Hz$), which reveals a direct equivalence to $1/h$.

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Bridging SI Units and Natural Units

The Natural Unit Framework

Natural units are a system where fundamental constants are normalized (often set to 1) to simplify physical laws. In contrast, our SI system uses separate units for mass ($kg$), temperature ($K$), length ($m$), and so on. To bridge these two perspectives, scaling factors are introduced. For example, when converting between energy and mass, the factor $c^2$ (from $E = mc^2$) serves as the natural conversion.

Within this framework, similar scaling factors are defined:

• $C_{kg}$ converts from coulombs to kilograms and is set to $1/c^2$ (kg/C) = 1V/$c^2$.
• $\text{kg}_{Hz}$ converts kilograms to natural frequency units it is found by isolating the  $c^2$/ h factor.

When these factors are multiplied, we obtain the natural unit scaling for charge:

$$C_{Hz}=C_{kg}\times {\text{kg}}_{Hz}$$

Unveiling the $1/h$ Factor

By plugging in the definitions, we find:

$$C_{Hz} = \frac{1}{c^2} \times \frac{c^2}{h} = \frac{1}{h}.$$

Numerically, since Planck’s constant is

$$h \approx 6.62607015 \times 10^{-34} \, \text{J·s},$$

its reciprocal is

$$\frac{1}{h} \approx 1.50919 \times 10^{33} \, \text{Hz/J}.$$

This equivalence is profound: it means that when you convert the SI coulomb into natural frequency units, the conversion factor is exactly $1/h$. In essence, the natural scaling of charge is set by the inverse of Planck’s constant.

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Implications of the $1/h$ Equivalence

A Deep Interconnection

This relation is more than a numerical curiosity—it reveals a deep unity in physics. Traditionally, constants like $h$ have been seen as fixed conversion factors. Yet, in this framework, they emerge as the bridges that connect diverse physical quantities. Here’s why that matters:

• Unified View of Physical Quantities:
  The fact that charge, mass, and energy are interrelated through factors like $c^2$ and $h$ suggests that these properties are merely different aspects of a single underlying reality. When expressed in natural units, the distinctions between them blur, unveiling an elegant unity in nature.

• Historical Perspective on Unit Definition:
  It is possible that, historically, the scaling of electrical constants was chosen in such a way that $h$ played a central role in setting these scales. If the scaling was intentionally or even inadvertently defined using $h$, then the appearance of $1/h$ as the conversion factor from coulombs to natural frequency units is hardly accidental. The enormous value of $1/h$ reflects the deep interconnections between the SI system and the natural, intrinsic scales of the universe.

From Human Conventions to Natural Realities

Our SI units are human-made constructs, optimized for practical measurements. However, when we express these units in terms of natural scales, we begin to see that what might appear as arbitrary or extrinsic is, in fact, deeply connected to the fabric of physical laws. The conversion $C_{Hz} = 1/h$ (Hz/C) is a perfect example: a human-defined unit (the coulomb) transforms into a natural unit of frequency with a conversion factor that is dictated by one of nature’s most fundamental constants.

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Conclusion

The discovery that converting coulombs to natural frequency units yields exactly $1/h$ is a striking insight. It underscores the fact that fundamental constants are not arbitrary numbers but are deeply embedded in the structure of physical laws. This equivalence not only bridges the gap between our human-scale SI system and a more intrinsic natural unit framework but also hints at a deeper unity among mass, energy, and charge. Such insights pave the way for a more unified understanding of the universe—one where the interplay of natural constants reveals the elegant simplicity underlying the apparent complexity of the world around us.