#superposition principle

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.

I. Charge Configurations; Target and Source Charges

We previously motivated Coulomb's Law, the force law between a pair of point charges in $3$-space. Namely, suppose we have a point charge $q^\prime$ at $\vec{r}^{\text{ }\prime}$ and a point charge $q$ at $\vec{r}$, so that they are spatially separated by $\vec{R}\equiv \vec{r}-\vec{r}^{\text{ }\prime}$. Then the target charge $q$ will experience a force $\vec{F}_{q^\prime\rightarrow q}$ as a result of the source charge $q^\prime$, and that force equals,

Recall that $u$ is some constant that ensures we end up with units of force. I’m choosing to label source charges with primed symbols and target charges with unprimed symbols, enabling the mnemonic “primed quantities produce effects.”

Of course, which charge we call a source charge versus a target charge is a matter of perspective because really each charge produces a photon cloud that the other interacts with. The “source” versus “target” nomenclature only makes sense when we’re calculating something with respect to a specific charge’s experience.

While point charges are great, they’re limited in their usefulness. We’d like to generalize electric properties to more complicated arrangements of charge. A generic distribution of charge throughout space is called a charge configuration.

There’s no need to jump straight to the most generic charge configurations; let’s start with simpler examples before we fully extend our machinery. The simplest charge configuration (beyond empty space or a single point charge) is a finite collection of point charges:

Specifically, suppose we have $N$ point charges: $q_1^\prime$ at $\vec{r}^{\text{ }\prime}_1$, $q_2^\prime$ at $\vec{r}^{\text{ }\prime}_2$, and so-on. As a shorthand, we’ll call this entire configuration $Q^\prime$, so that $Q^\prime = \{q_1^\prime,\cdots,q^\prime_N\}$. Next, we introduce a target charge $q$ at $\vec{r}$:

Note that the separation from $q$ is different for each charge in $Q^\prime$, so we need to index their separation vectors: label the separation vector from $q^\prime_i$ to $q$ as $\vec{R}_i$, such that $\vec{R}_i \equiv \vec{r}-\vec{r}^{\text{ }\prime}_i$.

We want to answer the following question: what force $\vec{F}_{Q^\prime\rightarrow q}(\vec{r})$ does the charge configuration $Q^\prime$ exert on the point charge $q$?

II. The Superposition Principle and Electric Fields

The key to determining $\vec{F}_{Q^\prime\rightarrow q}(\vec{r})$ lies in the antisocial behavior of photons. By definition, photons only interact with charged objects, and photons are uncharged. Photons pass right through each other!

For simplicity, suppose $Q^\prime$ consists of only two source charges, $q_1^\prime$ and $q_2^\prime$. Both source charges possess their own photon clouds, but these clouds don’t mind sharing the same space because photons don’t interact with each other. So wherever we place our target charge, there’ll be photons originating from both charges for it to absorb. But because those photons are coming uninterrupted from their sources, their combined effect is the same as the effects due to the individual charges added together. This leads us to the superposition principle of electrostatics.

The superposition principle mathematically states that the net force experienced by a charge $q$ due to a charge configuration $Q^\prime=\{q_i^\prime\}$ equals the sum of the forces that $q$ would experience due to each charge $q_i^\prime$ individually. Notationally, we write

But we know how to write the force between two point charges, so we may express this more explicitly as,

Because the target charge $q$ is independent of the source charge index $i$, we can factor it out of the sum.

This factorization has a physical interpretation that goes all the way back to charge being the capacity for an object to absorb or emit photons. When we place $q$ in the presence of the configuration $Q^\prime$, it’s going to absorb photons from $Q^\prime$ at a rate proportional to $q$. Meanwhile, the contribution in square brackets tells us about the photons $Q^\prime$ is generating, and is independent of $q$. In this way, we disentangle the force into a purely $q$-dependent factor and a purely $Q^\prime$-dependent factor, such that,

The target charge factor is the familiar electric charge of $q$, while the source charge factor is what we call the electric field $\vec{E}_{Q^\prime}$ due to $Q^\prime$:

We sometimes refer to an electric field as an $\vec{E}$-field for short. Because the $\vec{E}$-field is related to the density of source photons and that density varies throughout space, its value depends on the position $\vec{r}$ of the target charge. From its definition, we note that

To better facilitate building an intuition about the electric field, let’s focus on the electric field of a single point charge $q^\prime$, labeled $\vec{E}_{q^\prime}$ and equal to,

We first note that the electric field will point away from a positive source charge and inward for a negative source charge:

This is precisely the direction in which a positive target charge $q$ would experience a force!

In general, the $\vec{E}$-field generated by a charge distribution points in the direction that a positive point charge would experience a force.

Meanwhile, negative charges are pushed against the electric field.

If we imagine the electric field maps the flow of a fluid throughout space, then positive charges are pushed down the electric field flow, while negative charges are pushed up the electric field flow.

III. Electric Field versus Photon Cloud Density

For a single particle, the electric force and field strengths are proportional to the photon cloud density. This is not true of the electric field created by multiple charges.

As an extreme example, consider placing a positive target charge $q$ halfway between two positive source charges $q^\prime_1$ and $q^\prime_2$ of equal charge strength: $q^\prime = q^\prime_1 = q^\prime_2$. Call this halfway point $\vec{r}=\vec{r}_{1/2}$.

The number of photons at $\vec{r}=\vec{r}_{1/2}$ is double the number of photons due to either $q^\prime_1$ or $q^\prime_2$ alone. Simultaneously, the electric field $\vec{E}_{q_1^\prime,q_2^\prime}$ at $\vec{r}_{ 1/2 }$ due to this configuration equals

We can simplify this expression given the available geometric information. Because $\vec{r}_{ 1/2 }$ lies exactly halfway between $\vec{r}^{\text{ }\prime}_1$ and $\vec{r}^{\text{ }\prime}_2$ (see the illustration), $\vec{R}_1$ is exactly opposite $\vec{R}_2$:

Hence, the net electric field at $\vec{r}=\vec{r}_{ 1/2 }$ is,

Despite the existence of many photons at this point, the electric field vanishes! Following this through to the force, we find the force also vanishes:

We interpret this physically as a sort of stalemate between the charges: photons from $q_1^\prime$ are trying to push $q$ towards $q_2^\prime$, but photons from $q_2^\prime$ are pushing $q$ just as hard towards $q_1^\prime$. We’re forced to conclude that electric field strength is not correlated with photon cloud density for generic charge configurations.

IV. The Decomposition Corollary

The mathematics of classical E&M allow us to perform a trick: if we so desire, we can break a single source charge $q^\prime$ at a point $\vec{r}^{\text{ }\prime}$ into many source charges $q_i^\prime$ at $\vec{r}^{\text{ }\prime}$ so long as we’re careful to retain the same total amount of charge:

Because these all live at the same point of space, their separation vectors $\vec{R}_i$ are identical ($\vec{R}_i = \vec{R} = \vec{r}-\vec{r}^{\text{ }\prime} $) and the net electric field due to them equals,

which is precisely the electric field of the original source charge $q^\prime$! In this sense, charge is locally additive.

This is another consequence of photons being unable to see each other: given a charge emitting some number of photons, we may partition those photons into several groups. We can then allocate each group an appropriate fraction of charge and thereby exactly replicate the effects of that group. We can even (mathematically) pull photons out of thin air so long as we’re careful to introduce additional photons that exactly cancel their effects. From the perspective of the charges that are producing these photons, this is automatically ensured if our decomposition has the same total charge as our original source charge.

This result is essentially an additional facet of the superposition principle. We’ll refer to this property as the decomposition corollary:

The decomposition corollary states that a target charge will be identically affected by any two source charge configurations that have point-for-point identical total charges.

The decomposition corollary allows us to (locally) reorganize charges at our convenience.

It’s important to note that we can apply the decomposition corollary to empty space. Because it neither absorbs nor emits photons, we can think of empty space at a point $\vec{r}^{\text{ }\prime}$ as a charge-without-charge, $q^\prime_{empty} =0$. As such, empty space cannot exert a force on target charges. This effect is indistinguishable from instead having a positive charge $+q^\prime$ and a negative charge $-q^\prime$ both at $\vec{r}^{\text{ }\prime}$. Because any forces that $+q^\prime$ would cause will be exactly cancelled by the forces caused by $-q^\prime$, it’s equivalent to empty space as far as target charges are concerned!

This concludes our first generalization towards the full classical electromagnetic theory. Today we handled how a target point charge $q$ is pushed and pulled by a collection of source point charges $Q^\prime =\{q_i^\prime\}$. Now that we know how to handle forces generated by generic zero-dimensional charge configurations, we’ll expand to higher dimensions. Next time on SineOfPsi: we’re looking at line charges, surface charges, and volume charges.