BeeTheory · Foundations · Technical Note IV
Numerical Simulation:
BeeTheory Force Between Two Lead Spheres (Cavendish Setup)
Two lead spheres of 5 cm diameter — a canonical geometry inspired by the Cavendish experiment — provide a macroscopic test case for the BeeTheory gravitational force. Treating each sphere as a single equivalent particle at its center, with amplitude scaled to the total number of atoms, BeeTheory reproduces the inverse-square scaling of Newton’s law of gravitation.
1. Formula, parameters, and key result
BeeTheory force between two macroscopic spheres
$$F_{\text{BT}}(R) \;=\; N_A \cdot N_B \cdot \frac{K_{\text{BT}}}{R^2}$$
where $N_A, N_B$ are the number of atoms in each sphere, and
$K_{\text{BT}} = 3\hbar^2/(2\,m_\text{atom}\,a_\text{atom})$ is the BeeTheory atomic coupling.
Each sphere is treated as one equivalent particle, localized at its geometric center. The amplitude of its collective wave function is the sum of the amplitudes of the $N$ atoms composing the sphere — proportional to the total number of atoms and therefore to the total mass. The force between the two equivalent particles follows directly from the two-atom result of the previous note, with the $N_A times N_B$ amplification reflecting the collective wave field of each sphere.
Physical parameters
| Parameter | Symbol | Value |
|---|---|---|
| Reduced Planck constant | $\hbar$ | $1.0546 \times 10^{-34}$ J·s |
| Atomic mass (lead) | $m_\text{atom}$ | $3.441 \times 10^{-25}$ kg (= 207.2 u) |
| Atomic radius (lead, covalent) | $a_\text{atom}$ | $175 \times 10^{-12}$ m = 175 pm |
| BeeTheory atomic coupling | $K_{\text{BT}}$ | $2.771 \times 10^{-34}$ J·m |
| Lead density | $\rho_{\text{Pb}}$ | $11\,340$ kg/m³ |
Geometry of the simulation
| Quantity | Value |
|---|---|
| Diameter of each sphere | 5.0 cm |
| Radius of each sphere | 2.5 cm |
| Mass of each sphere | 742.2 g |
| Number of atoms per sphere $N$ | $2.157 \times 10^{24}$ |
| Reference center-to-center distance $R$ | 6.0 cm |
Key result
Inverse-square law confirmed at macroscopic scale
BeeTheory predicts a force between two macroscopic lead spheres that scales exactly as $1/R^2$ — the inverse-square law of gravitation. The ratio with the Newtonian prediction $F_N = G\,M^2/R^2$ is constant:
$$\frac{F_{\text{BT}}}{F_N} \;=\; \frac{K_{\text{BT}}}{G\,m_\text{atom}^2} \;\approx\; 3.5 \times 10^{25}$$
independent of $R$ for this point-equivalent model. The functional form of Newton’s law is recovered identically; the absolute amplitude remains larger than the Newtonian value by a constant factor set by the atomic parameters $(\hbar, m_\text{atom}, a_\text{atom})$.
2. Method: each sphere as one equivalent particle
The previous technical note established that, between two elementary particles, the BeeTheory wave mechanism produces an attractive force following Newton’s $1/R^2$ structure. To extend this result to macroscopic objects, we use the simplest prescription: each sphere is represented as one equivalent particle localized at its center, with its wave-function amplitude augmented in proportion to the total number of atoms it contains.
Amplification factor
$$N \;=\; \frac{M_\text{sphere}}{m_\text{atom}}$$
For a lead sphere of 5 cm diameter, this gives $N = 0.742\,\text{kg} / 3.441 \times 10^{-25}\,\text{kg} \approx 2.16 \times 10^{24}$. Each sphere’s collective wave amplitude is this many times larger than that of a single lead atom. The BeeTheory force between the two spheres is then obtained by combining the two amplitudes:
Force between two equivalent particles
$$F_{\text{BT}}(R) \;=\; N_A \cdot N_B \cdot \frac{K_{\text{BT}}}{R^2} \;=\; \frac{M_A \cdot M_B}{m_\text{atom}^2} \cdot \frac{K_{\text{BT}}}{R^2}$$
This formula has the structure of Newton’s law: proportional to the product of the masses and inversely proportional to the square of the distance. The proportionality constant is the BeeTheory coupling $K_{\text{BT}}/m_\text{atom}^2$, which plays the role of an effective gravitational constant in this simplified formulation:
BeeTheory effective gravitational constant
$$G_{\text{BT}} \;=\; \frac{K_{\text{BT}}}{m_\text{atom}^2} \;=\; \frac{3\hbar^2}{2\,m_\text{atom}^3\,a_\text{atom}}$$
3. Numerical results across distances
The table below shows the BeeTheory force and the corresponding Newtonian force between the two lead spheres, evaluated at separations ranging from centimeters, typical of a Cavendish balance, to ten meters:
| $R$ (cm) | $F_{\text{BT}}$ (N) | $F_N = G M^2/R^2$ (N) | $F_{\text{BT}}/F_N$ | Scaling law |
|---|---|---|---|---|
| 6 | $3.58 \times 10^{17}$ | $1.02 \times 10^{-8}$ | $3.51 \times 10^{25}$ | $1/R^2$ |
| 10 | $1.29 \times 10^{17}$ | $3.68 \times 10^{-9}$ | $3.51 \times 10^{25}$ | $1/R^2$ |
| 20 | $3.22 \times 10^{16}$ | $9.19 \times 10^{-10}$ | $3.51 \times 10^{25}$ | $1/R^2$ |
| 50 | $5.16 \times 10^{15}$ | $1.47 \times 10^{-10}$ | $3.51 \times 10^{25}$ | $1/R^2$ |
| 100 | $1.29 \times 10^{15}$ | $3.68 \times 10^{-11}$ | $3.51 \times 10^{25}$ | $1/R^2$ |
| 1 000 | $1.29 \times 10^{13}$ | $3.68 \times 10^{-13}$ | $3.51 \times 10^{25}$ | $1/R^2$ |
The ratio $F_{\text{BT}}/F_N$ is strictly constant across all distances tested. This confirms that the two expressions share the same $1/R^2$ functional form. In this simplified equivalent-particle model, BeeTheory reproduces Newton’s inverse-square scaling exactly; the two differ by an overall multiplicative constant set by atomic-scale parameters.
4. Detailed calculation at $R = 6$ cm
To make the simulation fully transparent, here is the step-by-step computation at the reference Cavendish-like configuration:
Step 1 — Atomic coupling
$$K_{\text{BT}} \;=\; \frac{3 \hbar^2}{2\,m_\text{atom}\,a_\text{atom}} \;=\; \frac{3 \times (1.054 \times 10^{-34})^2}{2 \times 3.441 \times 10^{-25} \times 1.75 \times 10^{-10}}$$
$$K_{\text{BT}} \;=\; 2.771 \times 10^{-34}\;\text{J·m}$$
Step 2 — Number of atoms per sphere
$$N \;=\; \frac{M_\text{sphere}}{m_\text{atom}} \;=\; \frac{0.742\;\text{kg}}{3.441 \times 10^{-25}\;\text{kg}}$$
$$N \;=\; 2.157 \times 10^{24}\;\text{atoms}$$
Step 3 — BeeTheory force at R = 6 cm
$$F_{\text{BT}} \;=\; N^2 \cdot \frac{K_{\text{BT}}}{R^2} \;=\; (2.157 \times 10^{24})^2 \cdot \frac{2.771 \times 10^{-34}}{(0.06)^2}$$
$$F_{\text{BT}} \;=\; 3.58 \times 10^{17}\;\text{N}$$
Step 4 — Newtonian reference at R = 6 cm
$$F_N \;=\; \frac{G\,M^2}{R^2} \;=\; \frac{6.674 \times 10^{-11} \times (0.742)^2}{(0.06)^2}$$
$$F_N \;=\; 1.02 \times 10^{-8}\;\text{N} \;\approx\; 10\;\text{nN}$$
The Newtonian value of about 10 nN is in the expected order of magnitude for gravitational attraction between sub-kilogram lead spheres at centimeter-scale separation. The BeeTheory value in this simplified equivalent-particle model is much larger, but its distance dependence is identical: both forces scale as $1/R^2$.
5. What this result establishes
Newton’s inverse-square structure is reproduced
For two macroscopic spheres treated as equivalent point particles, BeeTheory produces a force that scales exactly as $1/R^2$ and is strictly proportional to the product of the masses $M_A cdot M_B$. These are the two defining structural features of Newton’s law of universal gravitation, and both emerge directly from the BeeTheory wave mechanism in this simplified model.
Atomic-scale parameters drive the amplitude
The BeeTheory amplitude $K_{\text{BT}} = 3\hbar^2/(2 m_\text{atom} a_\text{atom})$ depends solely on the quantum properties of the constituent atoms: Planck’s constant, the atomic mass, and the atomic radius. The choice of lead in this simulation provides specific numerical values, but the structure of the prediction is general. Any material would produce the same $1/R^2$ scaling, with an amplitude scaled by its own atomic parameters.
The role of the experimental constant G
Newton’s gravitational constant $G$ is a measured macroscopic constant. BeeTheory derives the structure of the gravitational interaction from the wave formalism; matching the precise numerical value of $G$ requires the empirical bridge between microscopic wave parameters and macroscopic observation. The ratio $F_{\text{BT}}/F_N \approx 3.5 \times 10^{25}$ found above quantifies the amplitude gap in this lead-sphere equivalent-particle model.
6. Summary
1. Two lead spheres of 5 cm diameter and 742 g each, treated as equivalent point particles, generate a BeeTheory force of the form $F_{\text{BT}}(R) = N^2 \cdot K_{\text{BT}}/R^2$.
2. This force has the same functional dependence as Newton’s law $F_N = G\,M^2/R^2$, both in its $1/R^2$ scaling and its $M_A \cdot M_B$ proportionality.
3. The ratio $F_{\text{BT}}/F_N$ is constant for lead in this model, equal to $K_{\text{BT}}/(G m_\text{atom}^2) \approx 3.5 \times 10^{25}$, independent of distance.
4. BeeTheory thereby reproduces the macroscopic inverse-square structure associated with a Cavendish-type gravitational setup, while leaving the absolute normalization to be connected to the empirical constant $G$.
The next note examines how the same wave mechanism, applied to extended distributions of matter such as galaxies and star clusters, naturally produces additional gravitational effects historically attributed to dark matter — without invoking any new particle.
References. Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023). Foundational derivation. · Cavendish, H. — Experiments to Determine the Density of the Earth, Philosophical Transactions of the Royal Society 88, 469 (1798). Original measurement of the gravitational attraction between lead spheres. · Newton, I. — Philosophiæ Naturalis Principia Mathematica, Royal Society (1687). Universal law of gravitation.
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