Classical E&M III: The Electric Ambiance of Continuous Distributions
Multidimensional objects are more complex than points, but there is a method of relating the two. We can break multidimensional objects into tiny pieces that are qualitatively point-like. As the pieces become infinitesimally small, this relationship becomes exact:
In this way, multidimensional objects are translated into collections of points. We know the electric properties of point charges, and through that knowledge we'll derive the electric properties of generic charge configurations.
I. The Electric Field of a Line Charge, Infinitesimals
Consider a curve $C^\prime$ in $3$-space with total length $\ell^\prime$:
To begin, let’s assume this curve is uniformly charged.
“Charged” means it emits photons, whereas
“Uniformly” means it possesses only one type of charge (positive or negative) and each point of $C^\prime$ is equally likely to emit photons
For the time being, we restrict ourselves to positive-type charges. This means we don’t have to worry about positive and negative charges cancelling each other out. $C^\prime$ as a whole will be emitting photons at a certain rate $r_{C^\prime}$ proportional to its total charge $Q^\prime$.
Our intent--as mentioned in the intro--is to chop $C^\prime$ into tiny pieces and approximate each piece as a point charge. Thus, we partition $C^\prime$ into some large number $N^\prime$ of equal-length pieces:
Each partition piece has length $\Delta \ell_i^\prime = \ell^\prime /N^\prime$. In order to reproduce the total rate of photon emission (and because the charge is distributed uniformly over the curve), each piece needs to provide an emission rate equal to $r_{C^\prime}/N^\prime$. This implies a curious result: as $N^\prime$ gets larger, the number of pieces increase, and every piece contributes less to the total emission rate. Because emission rate is proportional to total charge (because there’s no positive/negative cancellations), any individual piece possesses less charge. In the limit that $N^\prime\rightarrow \infty$, the curve becomes equivalent to infinitely-many point charges, where each point has no charge!
Charge is evidently an inappropriate local quantity for a uniformly charged curve. This presents a roadblock on our path to continuous charge configurations. We can circumvent it by finding an alternative local quantity to work with. As usual, the trick lies in the problem: charge vanishes as the length of our partition pieces go to zero, so instead of considering charge outright let’s consider the linear charge density $\lambda_i^\prime$ per piece:
This has units of charge per length. For a uniformly charged curve cut into $N^\prime$ equal-length pieces, each piece has length $\Delta \ell^\prime_i = \ell^\prime /N^\prime$ and charge $\Delta q^\prime_i = Q^\prime /N^\prime$. Therefore,
where $\lambda_{C^\prime}$ is the total linear charge density of $C^\prime$, a finite result! Linear charge density let’s us rewrite the charge of each partition piece as,
We found this expression by considering a uniformly-charged, positively-charged curve, but--like we did when developing the integral calculus--we can reweight the charge contribution of each piece, so that $\lambda^\prime_i$ is a function along $C^\prime$. In doing so, we can access nonuniform line charges which possess differing amounts of (positive and negative) charge along their lengths.
Specifically, as the partition becomes infinitely fine, each point $\vec{r}^{\text{ }\prime}$ along the curve can be uniquely identified with an infinitely-short piece of curve possessing a certain linear charge density. The linear charge density thereby becomes a real-valued function $\lambda^\prime(\vec{r}^{\text{ }\prime})$ along $C^\prime$.
Now is as good as any time to introduce infinitesimal quantities. As $N^\prime \rightarrow \infty$, the charge of each piece technically vanishes ($\Delta q^\prime_i\rightarrow 0$). However, we can imagine “stopping short” of infinity and--in doing so--obtain a point-by-point identification of curve pieces yet somehow without fully removing the charge of each piece. If this seems too good to be true, it’s because it is. We must be careful to finish off that $N^\prime\rightarrow \infty$ limit before calling our calculation complete.
Warning: Infinitesimals are used widely throughout physics, oftentimes when juggling multiple limits. For example, sometimes we’ll break time into infinitesimal intervals while simultaneously breaking shapes into infinitesimal pieces. These are technically two different limits, so mixing them can cause trouble (recall how cautious we had to be when manipulating the double-limit in the Dirac delta post). Exercise caution when encountering infinitesimals in the wild.
It’s in this spirit that we talk about the infinitesimal charge $dq^\prime$ and the infinitesimal length $d\ell^\prime$ of each curve piece, which are related through the line charge density function $\lambda^\prime(\vec{r}^{\text{ }\prime})$ along $C^\prime$:
Hence, the infinitesimal limit associates every point $\vec{r}^{\text{ }\prime}$ with a point charge of strength $dq^\prime(\vec{r}^{\text{ }\prime})$. Of course, point charges produce electric fields, and $dq^\prime(\vec{r}^{\text{ }\prime})$ is no exception:
Because the charge is infinitesimally small, the electric field is also infinitesimally small. This encourages us to write the electric field with the lowercase $d$ we associated with other infinitesimals, yielding the label $d\vec{E}$. We also change the subscript on $d\vec{E}$ to reflect our intent to add up all of the electric field contributions along the curve:
We can rewrite this as a line integral contribution via the line charge density. In this case, an individual contribution becomes,
Note that this is a vector-valued object. In our integral calculus series, we discussed integrals of real-valued functions, but not vector-valued functions. How do we proceed?
II. Integrating Vector-Valued Functions
It turns out integration of vector-valued functions follows without much trouble if we assume that linearity of integrals plays nice with $3$-space vector operations.
Particularly, any element of $3$-space can be uniquely associated with three real numbers, called rectangular coordinates:
These coordinates are useful because they allow us to write vector addition and scalar multiplication with ease. Given two $3$-vectors $\vec{r}_1\equiv (x_1,y_1,z_1)$ and $\vec{r}_2\equiv (x_2,y_2,z_2)$, we can use vector addition to build a new $3$-vector labeled $\vec{r}_1+\vec{r}_2$, defined as,
Furthermore, given a real number $\alpha$, we can use scalar multiplication to transform $\vec{r}=(x,y,z)$ into a $3$-vector $\alpha\vec{r}$ defined as,
We will assume that integrals behave nicely with these properties, and treat the unit vectors $\hat{x}$, $\hat{y}$, $\hat{z}$ as constants. (This is true of our rectangular coordinates; there exist other coordinate systems of $3$-space which aren’t so lucky.)
Given a function $\vec{f}$ that maps $3$-vectors to $3$-vectors, we may decompose $\vec{f}$ into components,
Then we define the integral over a vector-valued function $f$ as a component-by-component operation, like so:
Although I’ve written this as a volume integral, the sentiment holds equally true for surface integrals and line integrals.
With this definition in mind, we can now add up the electric field contributions due to all the point charges along $C^\prime$, so that the electric field at $\vec{r}$ due to a line charge $\lambda^\prime$ equals,
We then find the force experienced by a target charge $q$ at $\vec{r}$ due to $\lambda^\prime$ by multiplying $\vec{E}_{C^\prime}(\vec{r})$ by $q$, so that we find,
III. The Electric Field of a Surface Charge
Let’s move on to the equivalent analysis of a surface charge. Again, by evenly distributing a finite amount of positive charge $Q^\prime$ over a surface $S^\prime$ we ensure that any point of $S^\prime$ contains only infinitesimal amounts of charge, thereby eliminating charge as a useful local metric of electric behavior.
Instead, a more appropriate local measure is the surface charge density $\sigma^\prime$. Its definition proceeds just like the line charge density: we divide $S^\prime$ into infinitely-many infinitely-small two-dimensional regions, so that each point $\vec{r}^{\text{ }\prime}$ on $S^\prime$ is uniquely identified with its own infinitesimal region. If the infinitesimal piece at $\vec{r}^{\text{ }\prime}$ has area $dA^\prime$ and charge $dq^\prime$, then the surface charge density $\sigma^\prime$ at $\vec{r}^{\text{ }\prime}$ is defined as,
such that it has units of charge per area. This allows us to transform the charged surface $S^\prime$ into point charges, each contributing an electric field amount equal to,
which combine to yield the electric field at $\vec{r}$ due to a surface charge $\sigma^\prime$:
as well as the force experienced by a target charge $q$ at $\vec{r}$ due to $\sigma^\prime$:
IV. The Electric Field of a Volume Charge
Finally, we distribute charge over a volume $V^\prime$. The appropriate local measure of charge is volume charge density, labelled $\rho^\prime(\vec{r}^{\text{ }\prime})$. To define volume charge density, we break $V^\prime$ into infinitesimals, each possessing some volume $d\tau^\prime$ and some charge $dq^\prime$. Subsequently, $\rho^\prime(\vec{r}^{\text{ }\prime})$ is defined as the multiplicative factor between the two.
Volume charge density has units of charge per volume. With this definition in place, each point charge contributes some infinitesimal amount of electric field,
which collectively generate the electric field at $\vec{r}$ due to a volume charge $\rho^\prime$:
and enable us to write the force experienced by a target charge $q$ at $\vec{r}$ due to $\rho^\prime$:
V. Combining Charge Densities into Generic Configurations
A fully generic charge density may be a combination of all of these objects. Suppose we have a charge configuration $CC^\prime$ made of point charges $q_{i_0}^\prime$ at $\vec{r}^{\text{ }\prime}_{i_0}$, line charges $\lambda^\prime_{i_1}$ on curves $C^\prime_{i_1}$, surface charges $\sigma^\prime_{i_2}$ on surfaces $S^\prime_{i_2}$, and volume charges $\rho^\prime_{i_3}$ on volume charges $V^\prime_{i_3}$. The electric field generated by $CC^\prime$ must be, by the principle of superposition,
If we place a point charge $q$ at a position $\vec{r}$, it will experience a force due to $CC^\prime$ equal to,
There we go: we now know how to calculate the electric properties of any distribution of charges throughout $3$-space. It’s worth noting that if we wanted, we could utilize Dirac deltas to express all charge densities as volume charge densities. Given a volume $V^\prime$ that encompasses a point charge $q^\prime$ at $\vec{r}^{\text{ }\prime}$, a line charge density $\lambda^\prime$ on $C^\prime$, and a surface charge density $\sigma^\prime$ on $S^\prime$, physicists will sometimes write,
Note that the units work out appropriately for these to be volume charge densities. However, these expressions are technically incorrect because Dirac deltas may only exist within integrals. Only in that context may these expressions be utilized without jeopardizing mathematical consistency.
And so, we’ve generalized the “source” aspect of electrostatics. Next week, we generalize our target charges and calculate the force between two generic charge configurations. See you then!
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