“Electrostatic force is that which governs the motion of the atoms. It is the force which causes them to collide and develop the life-sustaining energy of heat and light, and which causes them to aggregate in an infinite variety of ways, according to Nature’s fanciful designs, and forms all these wondrous structures we see around us. It is, in fact, if our present views be true, the most important force for us to consider in Nature.”
--Nikola Tesla
“Tesla, Marvel Of The Future.” Brooklyn Citizen, August 22, 1897.
Our goal today is to determine the force between two generic charge configurations. Last week we handled a generic source charge configuration by dissolving it into tiny point-like pieces, then adding up the contributions due to every piece. We can handle the target charge configuration identically.
This post is a stepping stone on our path to completing electrostatics. We’re getting there!
I. The Electrostatic Force between Volume Charge Densities
It helps that we already know the force a target point charge $q$ at $\vec{r}$ experiences due to an entire source charge configuration $CC^\prime$. Namely,
where the electric field due to $CC^\prime$ is the appropriate superposition of volume, surface, line, and point charges, each providing an electric field according to the following equations:
Suppose (for the sake of having a concrete example) we’re interested in the electric force between two volume charge configurations: $\rho^\prime$ occupying a volume $V^\prime$, and $\rho$ occupying a volume $V$. The electric field at $\vec{r}$ due to $\rho^\prime$ is given by $\vec{E}_{\rho^\prime}(\vec{r})$ above.
We dissolve $\rho$ into tiny point-like pieces, each possessing an infinitesimal amount of charge described by its volume charge density:
Because each $dq$ can only receive infinitesimal numbers of photons, the force on any $dq$ is also infinitesimal. We write this infinitesimal force as,
or, more explicitly,
Note that $\vec{r}$ labels which point-like piece of $V$ we’re talking about--it’s a variable that indexes our disintegration of the target $\rho$. Similarly, $\vec{r}^{\text{ }\prime}$ indexes the disintegration of the source $\rho^\prime$. As a result, $dq(\vec{r})$ depends only on $\vec{r}$ and is constant with respect to the $\vec{r}^{\text{ }\prime}$ integration. In other words, the disintegration of $\rho$ is independent of the disintegration of $\rho^\prime$.
We use the target $\rho$ to rewrite the force experienced by $dq(\vec{r})$ in terms of $d\tau$:
Because the infinitesimal $d\tau$ is independent of $\vec{r}^{\text{ }\prime}$ in the same way that $dq(\vec{r})$ was independent of $\vec{r}^{\text{ }\prime}$, we may immediately integrate the $\vec{r}$-dependent function in square brackets with respect to $d\tau$.
Consequently, the electric force $\vec{F}_{\rho^\prime\rightarrow\rho}$ that a target $\rho$ experiences due to a source $\rho^\prime$ is,
Note that this quantity has no explicit positional dependence. All positions have been integrated away! Once we’ve specified our charge distributions, the electric force between them is locked in. This might seem contradictory to our expression for the force experienced by a point charge, which has an $\vec{r}$-dependence; however, specifying a point charge configuration technically requires fixing the point charge's position $\vec{r}$ (in the same way we'd specify the location and orientation of a volume charge), so there is no contradiction.
We also note that $\vec{F}_{\rho^\prime\rightarrow\rho}$ possesses a nice symmetry between $\rho$ and $\rho^\prime$. In fact, swapping primed and unprimed quantities yields the same result, but with a minus sign: $\vec{F}_{\rho^\prime\rightarrow\rho} = -\vec{F}_{\rho\rightarrow\rho^\prime}$. Even charge configurations obey Newton's Third Law. This implies that the net force $\rho$ experiences due to itself is zero: $\vec{F}_{\rho\rightarrow\rho} = -\vec{F}_{\rho\rightarrow\rho} = \vec{0}$
II. The Electrostatic Force between Generic Charge Configurations
This procedure is readily generalizable. Given a generic source $CC^\prime$, the electric forces due to $CC^\prime$ on a volume charge $\rho$ on $V$, surface charge $\sigma$ on $S$, line charge $\lambda$ on $C$, and point charge $q$ at $\vec{r}$ are, respectively,
Finally, we write the force between generic charge configurations. If a charge configuration $CC$ is composed of point charges $q_{i_0}$ at $\vec{r}_{i_0}$, line charges $\lambda_{i_1}$ on curves $C_{i_1}$, surface charges $\sigma_{i_2}$ on surfaces $S_{i_2}$, and volume charges $\rho_{i_3}$ on volume charges $V_{i_3}$, then superposition yields the force that a target charge configuration $CC$ experiences due a source charge configuration $CC^\prime$:
This machinery enables us to calculate the force between any pair of charge configurations. This has important implications when combined with our arbitrary division between sources and targets. While we like to think of photons from a source being absorbed by a target, a source is simultaneously also absorbing its own photons. Similarly, the target (despite its nomenclature) is throwing photons at the source and itself. All of these exchanges generate forces.
To be explicit: we can imagine taking a volume charge density $\rho$ on $V$ and breaking it into two pieces, $V_1$ and $V_2$. The $V_1$ piece will have a volume charge density $\rho_1$ and the $V_2$ piece will have a density $\rho_2$.
Because we can regard $\rho_1$ as a source and $\rho_2$ as a target (or vice-versa), these two pieces will be exerting forces on one-another and thereby wanting to move about. To ensure the charge configuration remains static (as to stay in the realm of electrostatics), we must apply additional external forces to cancel out all of the electrostatic forces.
This perhaps illustrates how limited the realm of electrostatics truly is: nature desires a description more expansive than electrostatics, even according to the rules of electrostatics alone! Our days in electrostatics are inherently numbered.
Next week, we’ll describe special idealized charge configurations called multipoles. Soon we’ll be discussing electrostatics in materials, several Maxwell equations, and the speed of light that will provide us the bridge away from electrostatics once and for all.
Thanks for reading today’s post! Follow sineofpsi.tumblr.com for new physics content every Friday. Have questions about anything we’ve talking about? Send me an ask. I’m wishing you the best!
Introduction
In physics, the inverse square law refers to any physical quantity or strength that is inversely proportional to the square of the distance from the source of that physical quantity. This law applies to diverse phenomena, from gravity and electric fields to light and sound.
Mathematical Formulation of an Inverse Square Law
Mathematically, the inverse square law is written as:
$ F…
Charles-Augustin de Coulomb was born on June 14, 1736. A French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force. The SI unit of electric charge, the coulomb, was named in his honor in 1880.
How To Use The Combination Of Electrospinning And Centrifugal Spinning Technology To Produce PC, Polycarbonate Nanofibers
The simultaneous use of electrostatic and centrifugal forces in the electrocentrifugal spinning process enhances the drawing force applied on the polymer jets during the fiber spinning. During electrocentrifugal spinning, a combination of centrifugal, electrostatic, gravitational, and auxiliary forces acts on the initial jet; if some of those forces are higher than the jet surface tension and viscous resistance, the jet is fully stretched out and spun into ultrafine fibers (Fig. 10.10). Because the electrostatic and centrifugal forces are combined in electrocentrifugal spinning, it enables the use of lower operating voltage and slower rotating speed compared with electrospinning and centrifugal spinning methods alone.
