Magnetic Dipole Field

$$B = \frac{\mu_0}{4\pi} \left[ \frac{3(\mathbf{m}\cdot\mathbf{r})\mathbf{r}}{r^5} - \frac{\mathbf{m}}{r^3} \right], \quad m = qd/2$$

Status: Proven.

This is the standard result for the magnetic field produced by a magnetic dipole moment $\mathbf{m}$, derived directly from the Biot-Savart law (or, equivalently, from Maxwell’s equations) in the far-field limit. It’s the same mathematical form real bar magnets, current loops, and atomic magnetic moments all produce at distance.

Real, independent confirmation: this is 19th-century classical electromagnetism, confirmed to extraordinary precision across an enormous range of technologies — from compass behavior to MRI machines to particle-accelerator magnets. Not a contested or novel result.

Used in: Paper 3, reused unmodified — PBT proposes a mechanical origin for what a magnetic field physically is (directional particle flux rather than a fundamental field), not a change to this equation.