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I'm trying to understand electric current. Some resources say that it is the flow of charge, and other resources say that it is the quantity of charge that passes through a cross-sectional area over a period of time. This confuses me because I'm not sure if it is a quantity (quantity of charge) or an action (the flow of charge). Can you please provide me with a definitive definition?

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    $\begingroup$ Better question to ask would be that how are both of these definitions equivalent $\endgroup$ Sep 30, 2021 at 22:22
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    $\begingroup$ "Flow of charge" and "quantity of charge ... over a period of time" both mean the same thing. $\endgroup$ Sep 30, 2021 at 22:30
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    $\begingroup$ Can you explain more why you think these terms are different? Flow means amount per unit time - the two terms say exactly the same thing... $\endgroup$ Oct 1, 2021 at 7:35

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Simply stated, current is just a flow of charge. If, however you want to measure and quantify the amount of current, the quantity of current is the amount of charge passing a point over a period of time.

The first statement defines current, while the second defines its measurement.

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    $\begingroup$ "passing a point" is definitionally hazardous. "passing through a surface" is less so. $\endgroup$ Oct 1, 2021 at 11:46
  • $\begingroup$ @EricTowers You are of course correct, but actual measurements in general occur at points. $\endgroup$
    – Bill Watts
    Oct 1, 2021 at 19:21
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    $\begingroup$ That's not so clear for current... See, for example, current clamp, where the current through a surface isotopic to a disk is what is measured. $\endgroup$ Oct 1, 2021 at 21:09
  • $\begingroup$ @EricTowers There are several ways to measure current. The NIST defines the ampere as "One ampere is the current in which one coulomb of charge travels across a given point in 1 second." At nist.gov/si-redefinition/ampere-introduction. Previously the ampere was defined by the force between infinite parallel wires. At one time the ampere was defined by the amount of silver deposited from a silver nitrate solution. $\endgroup$
    – Bill Watts
    Oct 2, 2021 at 23:54
  • $\begingroup$ That is not the current definition of the Ampere; it is a simplified description appearing in prose describing the 2019 redefinition. The current definition makes no attempt to detail what electric current is, merely stating that an Ampere is the electric current which satisfies 1 C = (1 A)(1 s) (where the Coulomp is defined using the 2019 version of the elementary charge and the second is still defined using the hyperfine transition of Cesium). See the BIPM for instance. $\endgroup$ Oct 3, 2021 at 23:13
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You can probably find various definitions so I am not sure if there is any single "definitive" definition of electric current.

But as an engineer, I use the definition from the reference handbook for the Fundamentals Exam for becoming a professional engineering in the U.S. as follows:

"Electric current $i(t)$ through a surface is defined as the rate of charge transport through that surface or

$$ i(t)=\frac{dq(t)}{dt}$$

which is a function of time $t$, since $q(t)$ denotes instantaneous charge.

A constant current $i(t)$ is written as $I$, and the vector current density in amperes/m$^2$ is defined as $\vec J$."

Hope this helps.

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  • $\begingroup$ Writing a formula adds nothing to the understanding. $\endgroup$ Oct 2, 2021 at 13:57
  • $\begingroup$ @Peter-ReinstateMonica I respectfully disagree. The OP asked for a definition which addressed his question of whether current is simply a "quantity of charge", or simply the "flow of charge". This definition, with the formula, shows that it is the rate of charge transport across a surface. $\endgroup$
    – Bob D
    Oct 2, 2021 at 18:54
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Your confussion regarding the terms is specific to the use of the terms in English. There is the phenomenon of charge flowing through a wire and the physical quantity that is used to measure this flow. The two are different objects and they could be named with two different words. Which is done in french, Italian, Romanian and maybe other languages that I don't know. So the phenomenon is the "electrical current" and the quantity representing the rate of flow of charge is the "intensity of the electrical current". A hint about this is the usual leter used to desigante it "I" and not "C" (from current). The "intensity of the current" can be shorted to just "current" and so it may produce confusion with the term used for the phenomenon. In English this is so common that even in physics textbooks they call the quantity "current" even though they mean the intensity of the current. In Romanian, for example, I learned about the "intensity of the current" in school and this is what we called it in the classroom, but people in electrical industries and trades call it "current" as well or even the equivalent of amperage, like in English we use voltage for potential difference. In French and Italian physics texts I can see that they also use the equivalent of "intensity of the current" so there is less confusion than in English, I suppose.

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  • $\begingroup$ Interesting observation. In German, in many cases we use the word for the property (current, distance, mass etc.) also for the amount of that property. English and German are similar enough that I can stick with English here, the examples work 1:1 in German: "What's the current, distance, mass, force?". As is often the case for native speakers I find that entirely normal -- but for other things it is utterly impossible: "What's the light, money, sound, butter" does not make sense, or means something different. We have to ask "what's the amount (or sometimes, level) of x". $\endgroup$ Oct 2, 2021 at 14:12
  • $\begingroup$ I was wondering about German. When I said that it is something specific to English I meant from the lanuages I understand more or less. German it's on my list of languages I want to learn but did not get there yet. Thank you for your comment. $\endgroup$
    – nasu
    Oct 2, 2021 at 18:37
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Bob D and SomeUser, respectively, explained that current is the movement of charge per unit time, $$ I = \frac{\partial Q_\Omega}{\partial t},$$ and the integral of current density over cross sectional area, $$ I = \int_{\partial\Omega} \text{d}\mathbf{S} \cdot \mathbf{J}.$$ I will add that these two quantities are the same because of charge conservation. Indeed, by standard arguments (see Griffiths, Electrodynamics), we can write down the following equation $$\frac{\partial \rho}{\partial t} + \nabla\cdot \mathbf{J} = 0.\qquad\qquad(*)$$ The first term describes the change of charge density over time in some region $\Omega$, and the second describes the outward flow of the current density in that same region. Indeed, we see the quantities above once we integrate over the region $\Omega$, $$\int \text{d}V ~\frac{\partial \rho}{\partial t} = \frac{\partial Q}{\partial t}, \quad\text{and} \quad \int_\Omega \text{d}V~ \nabla \cdot \mathbf{J} = \int_{\partial\Omega}\text{d}\mathbf{S}\cdot \mathbf{J},$$ where we've used the divergence theorem for the second equality.

Since charge density and current density are not in general separately conserved, we can define a new quantity, called the four-current, which is conserved by the equation $(*)$: $$J^\mu = (\rho, \mathbf{J}),\qquad \partial_\mu J^\mu = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0.$$ The statement $\partial_\mu J^\mu= 0$ is equivalent to the conservation of charge. See here for more.

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Electric current $I$ through a cross section of a conductor is: $$\lim_{\Delta t \to 0} \frac{\Delta q}{\Delta t}$$ where $\Delta q$ is the amount of charge that passed through that cross section during the period of time of length $\Delta t$.

Therefore, it is a scalar quantity.

Alternatively, if the area of the cross section is $S$, the quantity of charge per unit volume of the conductor is $n=\frac{N}{V}$ ($N$ is the number of charged particles in volume $V$) and $v_d$ the average drift velocity in the direction of the electric field, then the amount of charge that passes through that cross section in time $\Delta t$ is $\Delta q=q_0 \times n \times S \times v_d \times \Delta t$ ($q_0$ is the elementary charge of a single particle).

This is because all the charge that passes through during this period is located within a cylinder of height $v_d \Delta t$ and volume $V=Sv_d \Delta t$ which contains exactly $\Delta N=nV=nSv_d \Delta t$ charged particles.

From this, the current $I_S$ through this cross section would be $$\lim_{\Delta t \to 0} \frac{\Delta q}{\Delta t}=\lim_{\Delta t \to 0} \frac{q_0nSv_d \Delta t}{\Delta t}=\lim_{\Delta t \to 0} q_0nSv_d=q_0nSv_d$$.

All in all, it is a scalar quantity that describes the amount of charge that flows through a surface during a certain period. Its units are $\frac{Coulombs}{second} \equiv Amperes$.

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I prefer to define it as \begin{equation} I = \iint_{S} \mathbf{J} \cdot \, d\mathbf{S} \end{equation}

which can be rewritten as \begin{align} I &= \frac{\iint_{S} \rho \mathbf{v} dt \cdot \, d\mathbf{S}}{dt} \\ \end{align} The charge density is integrated over the surface and some thickness given by the velocity (actually, just the speed on the direction perpendicular to $S$) of the charges: if the charges are moving faster, further charges will reach the surface in the same time $dt$. You can see units match up. \begin{align} &= \frac{\iint_{S} \rho d\mathbf{r} \cdot \, d\mathbf{S}}{dt} \\ &= \frac{\iint_{S} \rho \,dr_{\perp} \,dS}{dt} \\ \end{align} Bassically, the numerator is the total amount of charge that will pass trough $S$ in $dt$. Thus, we can write it $dq$. \begin{equation} I = \frac{dq}{dt} \end{equation}

This do not intend to be a "proof" nor similar. I just wanted to point out that both concecpts are equivalent. Myself, I was introduced this concept the other way arround.

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