- #1
jcap
- 170
- 12
I was wondering if the [Feynman-Heaviside formula](http://www.feynmanlectures.caltech.edu/II_21.html) 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
$$
\begin{eqnarray}
\vec{R} &=& \vec{r_2}(t)-\vec{r_1}(t-R/c)\tag{2}\\
R &=& |\vec{R}|\notag\\
\vec{n} &=& \frac{\vec{R}}{R}\notag
\end{eqnarray}
$$
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 electric 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](https://en.wikipedia.org/wiki/Gravitoelectromagnetism) 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?
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
$$
\begin{eqnarray}
\vec{R} &=& \vec{r_2}(t)-\vec{r_1}(t-R/c)\tag{2}\\
R &=& |\vec{R}|\notag\\
\vec{n} &=& \frac{\vec{R}}{R}\notag
\end{eqnarray}
$$
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 electric 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](https://en.wikipedia.org/wiki/Gravitoelectromagnetism) 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?