Why is current defined as $dQ/dt$ even though it is not defined as the rate of 'change' of flow of charges?

Question

I do not understand this definition. $dQ/dt$ represents the rate of CHANGE of charge flow at an instant even though current is defined as only the charge flow per unit time.

Answer 1

Why is current defined as $\mathrm dQ/\mathrm dt$ [...]

It isn't. Given an ideal, infinitely thin wire and a choice of which direction to call positive, the current at some point $p$ is the rate at which charge flows past $p$. In other words, $I(t) \mathrm dt$ is the amount of charge which flows past $p$ in the interval $[t, t+ \mathrm dt]$. If your wire is not infinitely thin, then you may have to start talking about current density, but the spirit remains the same - neither the current nor the current density are defined to be the derivatives of anything.

With that being said, sometimes we consider situations in which the current happens to equal the derivative of some quantity. The simplest example is when we consider the current in a wire which is in series with a capacitor. Because the amount of charge $Q(t)$ on a capacitor is a well-defined quantity at every moment, and because electric charge is conserved, then - assuming that charge is not accumulating anywhere other than the capacitor - we must have that the rate of change of the charge on the capacitor is equal to the current which flows through the wire, i.e. $I(t) = Q'(t)$.

Answer 2

though current is defined as only the charge flow per unit time

Current, $i(t)$, is defined as the rate of charge transport through a surface., or

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

For a microscopic view of electric current, see

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/miccur.html

Hope this helps.

Answer 3

Take volume as an example.
If a tank has a hole in it and the volume of liquid in the tank is decreasing then $\frac{dV}{dt}$ is the instantaneous rate of change of volume of liquid in the tank.
However, if you are measuring the flow of liquid through the hole, then you might call $\frac{dV}{dt}$ the instantaneous rate of flow of liquid through the hole or volume flux.

In electricity the equivalent statements about charge, $\frac{dQ}{dt}$ might be, the rate of change of charge stored on the capacitor, or the rate of flow of charge through a wire connected to the capacitor (electric current).

So the wording can be changed according to the context.

Answer 4

Your $$ \dfrac{dQ}{dt} $$

literally means charge per unit time. That's the charge flow.

The rate of change of the charge $Q(t)$ that is passing through a surface which is perpendicular to the current density is your definition of current. The instantaneous rate of change of the charge flow would be equal to $$ \dfrac{d^2Q}{dt^2} $$

Answer 5

I think you are misunderstanding that in the definition of current, the charge $Q(t)$ is an accumulated quantity, not a property of some physical object. Although these concepts are closely related, they are not the same.

References to charge such as "the charge of the electron" or "the value of a charge distribution" do so in the context of charge as a physical property of matter. A function $Q(t)$ in this context would describe something like the value of an object's charge property over time.

This is to be distinguished with the concept of an accumulated quantity of charge. In this context the function $Q(t)$ refers to the combined charge of all matter which has flowed through a particular surface in space since the reference time $0$ to the given time $t$. This is intuitively equal to the time integral of the rate of charge flow through the surface: $$Q(t) = \int_{0}^t I(t')\mathrm dt'$$ The "rate of charge flow through the surface" $I(t)$ is what is defined as "the current through the surface" and this expression can be inverted using the fundamental theorem of calculus to express current as the rate of change of accumulated charge: $$I(t) = \frac{\mathrm d}{\mathrm dt}\int_{0}^t I(t')\mathrm dt' = \frac{\mathrm dQ(t)}{\mathrm dt}$$ So to reiterate, current is not defined as the rate of change of some property, it is the rate of change of an accumulated quantity.

With all that being said, a remaining source of ambiguity between these 2 concepts that needs to be addressed is the law of local charge conservation, which roughly says "The total charge in a region can only be changed by charged matter flowing through the boundary of the region", i.e. charge can not teleport into/out of the region. It relates the rate of change of charge (as a property) in a region to the charge flowing through boundary surface in the form of a continuity equation: $$\frac{\mathrm dQ_\mathrm{enc}}{\mathrm dt} = I_\mathrm{in}$$

Although this law is in the exact same form as the definition of current, it is merely a relation between the charge in a region and current flowing into the region. The difference is similar to how Newton's 2nd law $\mathbf F = m\mathbf a$ is not a definition of force or acceleration but rather a relation of force to the acceleration of an object.

Hope this clears it up

Answer 6

Calling current '$\frac{\text d Q}{\text d t}$' is uncontroversial when there is a $Q$ that has a definite value at any time $t$. An example would be the current into a capacitor plate, which is the rate of gain of charge by the plate, $Q$ being the charge on the plate at time $t$.

Use of the notation $\frac{\text d Q}{\text d t}$ is hard to justify when there is no definite $Q$. An example would be the current through a resistor connected to a battery. All the same, $\frac{\text d Q}{\text d t}$ is often used to mean current or rate of flow, and I've never known confusion to result.

I prefer $\dot Q$ for rate of flow, but I don't claim that this is any less bad than $\frac{\text d Q}{\text d t}$.