# Generalizing Coulomb’s Law

The forces between two electric charges in arbitrary motion are complicated by velocity, acceleration, and time-delay effects. The forces need not even lie along the line joining the two charges!

Suppose a source charge $q^\prime$ is at position $\vec r^{\,\prime}$ with velocity $\vec v^{\,\prime}$ and acceleration $\vec{a}^{\,\prime}$, and a test charge $q$ is at position $\vec r$ with velocity $\vec v$ and acceleration $\vec{a}$. Let their separation $\vec{\mathcal{R}}= \vec r – \vec r^{\,\prime}$. Since electromagnetic “news” travels at light speed $\color{blue}c\color{black} = 1$ (in natural units), define the earlier retarded time $s < t$  implicitly by $t – s = \mathcal{R}[s] / \color{blue}c\color{black} = \mathcal{R}_{\color{red}s}$.

Liénard-Wiechert electric and magnetic potentials at test charge $q$ due to source charge $q^\prime$ are

$$\varphi_{\color{red}t} =\frac{q^\prime_e}{4\pi \mathcal{R}_{\color{red}s} }=\frac{q^\prime}{4\pi \mathcal{R}_{\color{red}s} } \left( \frac{1}{1 – \hat{\mathcal{R} }_{\color{red}s} \cdot \vec v^{\,\prime}_{\color{red}s}} \right), \hspace{1cm} \vec A_{\color{red}t}= \varphi^\prime_{\color{red}s} \vec v^{\,\prime}_{\color{red}s},$$

where the source charge $q^\prime$ is “smeared” to an effective charge $q^\prime_e$ by its motion, and the subscripts indicate evaluation times. If the source charge velocity $\vec v^{\,\prime}_{\color{red}s} = 0$, then the potentials simplify to the electrostatics limit. Differentiate the potentials to find Maxwell’s electric and magnetic fields

$$\vec{\mathcal{E}} = – \vec \nabla \varphi – \partial_t \vec A \hspace{1cm} \vec{\mathcal{B}} = \vec \nabla \times \vec A,$$

and the Lorentz force law

$$\vec F = q \left( \vec{\mathcal E} + \vec v \times \vec{\mathcal B}\right)$$

generalizes Coulomb’s law to

$$\vec F_{\color{red}t} = q \frac{ q^\prime } { 4 \pi \mathcal{R}^2_{\color{red}s}} \left( \frac{ 1 } {1 – \hat{\mathcal{R}}_{\color{red}s} \cdot \vec{v}^{\,\prime}_{\color{red}s} }\right)^3 \bigg( \frac{}{} (1-v^{\prime 2}_{\color{red}s})(\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) + \vec{\mathcal{R}}_{\color{red}s} \times \left((\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) \times \vec{a}^{\,\prime}_{\color{red}s} \right)$$ $$\hspace{3.4cm} + \,\vec v_{\color{red}t} \times \left( \hat{\mathcal{R}}_{\color{red}s} \times \left( (1-v^{\prime 2}_{\color{red}s})(\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) + \vec{\mathcal{R}}_{\color{red}s} \times \left((\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) \times \vec{a}^{\,\prime}_{\color{red}s} \right) \right) \right) \, \bigg).$$

If the test charge velocity $\vec v = \vec 0$, then the magnetic terms vanish; if the source charge acceleration $\vec{a}^{\,\prime} = \vec 0$, then the radiation terms vanish; if, in addition, the source charge velocity $\vec v\,^\prime = \vec 0$, then the generalized Coulomb’s law reduces to the familiar electrostatic limit. Finally, Newton’s second law with Einstein’s momentum

$$\vec F = \frac{d\vec p}{dt} = \frac{d}{dt} \frac{ m \vec v }{\sqrt{1 – v^2}} = \frac{ m \vec a}{\sqrt{1 – v^2}} + \frac{ m \vec v}{(1 – v^2)^{3/2}} \vec v \cdot \vec a$$

generates the motion equations using 3-vectors and lab time (instead of 4-vectors and proper time), succinctly summarizing all of electromagnetism.

Using Mathematica, I eliminated present time $t$ from the motion equations with the substitution $t = s + \mathcal{R}_{\color{red}s}$ and numerically integrated them with respect to the retarded time $s$ for various initial conditions, as in the figure below. Positive and negative charge pairs tugged by non-zero net forces spiral into one another as they radiate away energy and momentum, graphically demonstrating the instability of classical atoms.