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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?

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