# Feynman-Heaviside formula and Mach's principle

I was wondering if the Feynman-Heaviside formula for the electric field of a moving charge could be used to write down the force/reaction force between charges $$q_1$$ and $$q_2$$ in a Machian purely relational way.

The retarded electric force $$\vec{F_{12}}$$, on a charge $$q_2$$ that is at rest at its current position $$\vec{r_2}(t)$$, due to a moving charge $$q_1$$ at its earlier position $$\vec{r_1}(t-R/c)$$ is $$\vec{F_{12}} = \frac{q_1 q_2}{4 \pi \epsilon_0} \left[\frac{\vec{n}}{R^2} + \frac{R}{c}\frac{d}{dt} \left(\frac{\vec{n}}{R^2}\right) + \frac{1}{c^2} \frac{d^2\vec{n}}{dt^2}\right]\tag{1}$$ Where $$\begin{eqnarray} \vec{R} &=& \vec{r_2}(t)-\vec{r_1}(t-R/c)\tag{2}\\ R &=& |\vec{R}|\\ \vec{n} &=& \frac{\vec{R}}{R} \end{eqnarray}$$

The expression for the force $$\vec{F_{12}}$$ in Eqn. $$(1)$$ is written entirely in terms of the magnitude and direction of the relative vector between the current position of charge $$q_2$$ and the earlier position of charge $$q_1$$. No variables defined in terms of an absolute reference frame are used.

The first two terms on the righthand side of Eqn. $$(1)$$ are the near-field Coulomb term and its correction that fall off like $$1/R^2$$ whereas the last is the far-field radiative term that falls of like $$1/R$$.

The advanced reaction force $$\vec{F_{21}}$$ back on charge $$q_1$$ at its earlier position $$\vec{r_1}(t-R/c)$$ due to charge $$q_2$$ at its current position $$\vec{r_2}(t)$$ is then just

$$\vec{F_{21}} = - \vec{F_{12}}\tag{3}$$

Thus Newton's third law of action and reaction is obeyed through an influence that travels at the speed of light forward in time from $$q_1$$ to $$q_2$$ and then backward in time from $$q_2$$ to $$q_1$$.

This reaction force provides an electromagnetic inertial force back on charge $$q_1$$ at the earlier time $$t-R/c$$ due to the presence of the charge $$q_2$$ at the current time $$t$$.

One could test for this electromagnetic inertia by accelerating an electron of charge $$-e$$ inside an insulating charged sphere of radius $$R$$ and charge $$Q$$. The electron’s inertia should be increased by an amount ~ $$eQ/(4 \pi \epsilon_0 c^2 R)$$.

Finally this Feynman-Heaviside force could be generalized to the weak-field gravitational case simply by substituting masses for charges and Newton’s constant $$G$$ for $$-1/(4 \pi \epsilon_0)$$. Thus standard inertia could be explained as the result of the gravitational advanced Feynman-Heaviside reaction force acting back on an accelerated test mass from all the other masses in the Universe.

Does this make sense?