Why don't electric current and volumetric flow rate express units/dimensions of area in their denominators?

Question

The definition of current is $I = \frac{dq}{dt}$ and the definition of volumetric flow rate is $Q = \frac{dV}{dt}$.

In written, non-mathematic language, I have seen current described as: "Electric current is defined as the rate at which charge flows through a surface (the cross section of a wire, for example)" (source: https://physics.info/electric-current/)

Similarly, I have seen volumetric flow rate described as: "The volume flow rate Q of a fluid is defined to be the volume of fluid that is passing through a given cross sectional area per unit time." (source: https://www.khanacademy.org/science/physics/fluids/fluid-dynamics/a/what-is-volume-flow-rate)

In both cases, we see that these concepts are established with reference to a cross sectional area.

Now, I recently learned about the concept of current density, which is the amount of charge per unit time that flows through a unit area of a chosen cross section. Although I have not seen an equivalent term in the context of fluid dynamics, one could easily imagine a similarly derived term (i.e. a "volumetric flow rate density").

Importantly, current density has the dimensional units of $\frac{I}{L^2}$.

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So my question is, "What exactly is going on here?". All terms seem to be described with respect to some cross sectional area, yet only the current density (and the "volumetric flow rate density" made-up term) carries the units of area in its denominator.

At first, I sort of thought that the cross sectional term in current density behaved like a percentage. i.e. the diameter of the wire that the current flows through in a circuit could be thought of as "the universe"...or $100 \text {%}$ of the area. If I wanted to know how much current flows through $50 \text{%}$ of the wire, I would divide that "universal current" by $2$. However, clearly this term is not behaving like a percentage because it has actual units!

Any insights would be greatly appreciated! Cheers~

Answer 1

We get many questions on here that ask, "why is this physical quantity defined in this way?" And the answer is always the same: because it's useful.

There are many instances where we don't really care about the actual area used to determine the current. For example, if someone has already determined the resistance of a resistor, and I want to know what the charge flow through that resistor will be when I apply some potential difference, I don't want to also have to worry about the cross-sectional area of that resistor. Just knowing how many charges that flow through it in some unit time is good enough.

Another example is in circuit components that are in series. The current is the same through each component, even though the current density will change depending on the cross-sectional area of each component.

I'm sure you can think of many other examples where current is sufficient and current density is not needed (just go through any introductory physics book and look for equations with $I$ in them). On the offer hand, there are cases where current density is important too. Both current and current density are important depending on the situation, so both have been defined and are widely used.

Answer 2

One might describe the terms as:

• Current is "the amount of charge passing through some chosen surface".
• Current density is "the amount of charge passing through some chosen surface divided by the area of that surface".

There's nothing inherent in the definition of current about the amount of surface area involved. When defining a current, you don't have to consider the surface area of the surface you choose, you just have to choose a surface.

Current is useful in something like idealized RL circuit analysis, where current density would contain extraneous information. Current density shows up in Ampère's circuital law and likely (although I'm not certain) in some aspects of heat dissipation.

Answer 3

Quite simply, current $I$ is current density $\rho$ integrated over a surface:

$$ I = \iint_S \vec{\rho} \cdot d \vec{S} $$

So when one speaks of the current $I$, the surface $S$ is implicit in the above definition. Typically the current density is very localized (ie a current carrying wire), which is why it’s not necessary to explicitly define the surface (it’s implied via the cross section of the wire). That convention might break down though in more complicated scenarios, such as a coaxial cable.

The integral for volumetric flow would look similar to the above integral, but probably would contain a divergence term for compressible fluids.