Differential charge existing

Question

We define current by $I=\frac{\mathrm{d}q}{\mathrm{d}t}$. Here, $\mathrm{d}q$ is the infinitesimal element of charge. But again,we know that charge is quantised meaning there is a finite value to the smallest amount of charge which is $e$. Since $\mathrm{d}q$ is infinitely small, $\mathrm{d}q<e$. Then how can $\mathrm{d}q$ charge even exist?

Answer 1

You're mixing up two descriptions that are, in practice, separate.

$i=dq/dt$ is usually used in macroscopic physics, when it is understood that you don't study actual individual electrons. In fact, most of the corresponding physics laws predate quantum mechanics, even predate the discovery of the electron.

In other words, whether $dq$ is the "small" charge contained in a "small" volume $dV$ or crossing a section during a "small" duration $dt$, it must be understood as containing a mesoscopic number of charge carriers (large compared to unity, small compared to $\mathcal{N}_A$).

In this context, reducing either $dV$ or $dt$ to the point that $dq$ contains only a few electrons makes you step outside the validity of those laws of physics.

Answer 2

The same holds true for water. I presume you don't have problem using $d V/dt$ for the flow of a volume $V$ of water. Yet we know that water is ultimately discrete molecules...

The point is: the discrete nature of the water molecule or the electric charge does not manifest itself much in everyday life so it's much more convenient to think of macroscopic amounts of water or charge as being continuous rather than discrete quantities.

Answer 3

I believe this is a nice example that there is a distinction between what a physical theory is and what the physical world is. To borrow Stephen Hawking's words,

I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation.

The point is not whether there actually exists a quantity $\mathrm{d}q$ of charge. Classical Electrodynamics is not concerned with such fundamental questions. The point is that in a myriad of situations, one often deals with charges that can vary by very tiny amounts (for example, of the order of $e$), and when compared with the total charges these quantities are extremely small. In these situations, one can approximate these small, albeit discrete, charge changes $\Delta q$ by continuous variations $\mathrm{d} q$. One then gets access to the tools of Calculus and a lot of extremely powerful machinery that makes the computations way easier.

What if the total charges are not that large, so we are actually dealing with, e.g., one or two electrons at a time?

In these weak field limits, quantum effects might start to kick in. What happens is that, at some point, the classical description will crumble down and fail. There simply are phenomena which are not described by Classical Electrodynamics, and instead one needs to work with Quantum Electrodynamics. This is not a problem: all known physical theories have such a limit. As Anthony Zee puts in Quantum Field Theory in a Nutshell,

If anyone tries to sell you a field theory claiming that it holds up to arbitrarily high energies, you should check to see if he sold used cars for a living.

In short, there is no amount of charge $\mathrm{d}q$ in the actual Universe (at least as far as present day knowledge can say). However, this doesn't matter in Classical Electrodynamics. Classical Electrodynamics is not concerned with a fundamental description of the Universe, but rather with an adequate description of a wide range of phenomena, which do not include quantum aspects, much less the fact that charge is quantized. This is not a problem: in fact, all theories we know have similar limitations. The reason for using $\mathrm{d}q$ is that it works, up to a controlled and reasonable amount of error. Once this error grows too large for you, you'll need a better description, such as Quantum Electrodynamics.

"All magnetic phenomena are NOT due to electric charges in motion"

A similar problem to the one you mentioned was recently raised by S. Fahy and C. O'Sullivan in the American Journal Physics 90, 7 (2022). They mention how many Classical Electrodynamics books state that magnetic phenomena are due to currents, when in fact a lot of the phenomena we experience are due to electron spin, which has nothing to do with electric charges in motions. D. J. Griffiths then answers their points on American Journal Physics 90, 9 (2022), having a position that I consider similar to the one I expressed for your problem: while it is true that magnetic phenomena are due to spin, intrinsic magnetic dipoles do not fit within what is commonly known as Classical Electrodynamics. I believe these short papers (both of which are open-access) might also interest anyone interested in this post, as they discuss what I consider essentially the same problem (the limits of Classical Electrodynamics), albeit in a slightly different setting.

Answer 4

The mistake you are making is interpreting $dq(t)$ in the definition of current as an infinitesimal amount of charge. In the definition of current $q(t)$ denotes the instantaneous charge crossing a surface. Consequently, current $i(t)=dq(t)/dt$ through a surface is defined as the rate of charge transport through that surface.

Electric current is measured in Coulombs per second and the Coulomb equals 6.24 x 10$^{18}$ elementary charges (protons or electrons). Since the smallest amount of charge through a surface at any instant is that of a single (indivisible) electron or proton, the current for a single elementary charge through a surface per second is 1.602 x 10$^{-19}$ Ampere.

Hope this helps.

Answer 5

Good question, so good that Purcell and Morin address it directly in their Electricity and Magnetism text.

On an atomic scale, of course, the charge density varies enormously from point to point; even so, it proves to be a useful concept in that domain. However, we shall use it mainly when we are dealing with large- scale systems, so large that a volume element $dv = dx dy dz $ can be quite small relative to the size of our system, although still large enough to contain many atoms or elementary charges.

I had in mind a more stirring quote, but this is the one I could track down. Maybe Griffiths or Jackson also comment on it. Anyway, suffice to say it is a good question. The usual answer is to work in the limit where things are small, but still big enough that we can still meaningfully think of continuous distributions of charge. Lucky for us this ends up not being such a delicate balance because of how dense with charged particles most materials are and how steady charge distributions are at low energies in most materials.

I could swear there is an exercise on exactly this in one of the classic texts that is meant to put your mind at ease, but I can't find it. Maybe someone else will know what I'm talking about.