KAHIBARO
Table of Contents
Microscopic Motion in a Current
When electric current flows through a metal wire, the charges inside the wire are already moving all the time. In a metal, many electrons move randomly in all directions because of their thermal motion. If there is no electric field, this motion has no preferred direction, so the average motion is zero and there is no net current.
When an electric field is applied across the wire, the electrons still keep their random motion, but now they also gain a small average motion in one direction. This average directed motion is called drift velocity.
Drift velocity is not the same as the random speed of individual electrons. The random thermal speeds are very large, but because they are in all directions, they cancel out on average. Drift velocity is much smaller, and it is the part of the motion that produces electric current.
Direction of Drift
In metallic conductors, the moving charges are electrons, which have negative charge. Because they are negatively charged, their drift is opposite to the direction of the electric field.
Conventional current is defined as the direction positive charges would move. This means that in a metal, the direction of current is opposite to the drift velocity of the electrons.
For example, if the current in a wire is to the right, the electrons drift on average to the left.
Important direction rule:
In a metal wire,
$ $$
\text{electron drift direction} = \text{opposite to current direction}
$$
and
$$
\text{electron drift direction} = \text{opposite to electric field direction}
$$
Definition of Drift Velocity
Drift velocity is usually written as $v_d$. It is the average velocity with which charge carriers move through a conductor because of an applied electric field.
If the charge carriers are electrons, then $v_d$ refers to the average electron drift speed and direction. In other materials, such as semiconductors or electrolytes, the charge carriers may be different, but the idea is the same.
A useful way to understand drift velocity is to imagine a crowded hallway. Each person moves in a complicated way, bumping into others, but the whole crowd may slowly shift in one direction. That slow average shifting is like drift velocity.
Relation Between Current and Drift Velocity
Consider a wire with cross sectional area $A$. Suppose the wire contains charge carriers of number density $n$, meaning there are $n$ carriers per unit volume. If each carrier has charge $q$ and drifts with speed $v_d$, then the current is
$$
I = nqAv_d
$$
This formula connects the microscopic picture of moving charge carriers to the macroscopic quantity current.
Here,
$ I $ is the current,
$ n $ is the number of charge carriers per unit volume,
$ q $ is the charge of each carrier,
$ A $ is the cross sectional area of the conductor,
$ v_d $ is the drift velocity.
If the carriers are electrons, then the magnitude of the charge is $e = 1.60 \times 10^{-19}\,\text{C}$. In current calculations, we often use the magnitude of the charge and handle direction separately.
Key drift velocity formula:
$$
I = nqAv_d
$$
So,
$$
v_d = \frac{I}{nqA}
$$
This shows that drift velocity is larger when current is larger, and smaller when the wire is thicker or has more charge carriers per unit volume.
How the Formula Arises
In a short time interval $\Delta t$, the charge carriers drift a distance
$$
\Delta x = v_d \Delta t
$$
The volume of charge passing through a cross section of the wire is
$$
\Delta V = A \Delta x = Av_d \Delta t
$$
If there are $n$ carriers per unit volume, then the number of carriers in that volume is
$$
n \Delta V = nAv_d \Delta t
$$
Since each carrier has charge $q$, the total charge passing through is
$$
\Delta Q = nqAv_d \Delta t
$$
Using the definition of current,
$$
I = \frac{\Delta Q}{\Delta t}
$$
we get
$$
I = nqAv_d
$$
This derivation shows that drift velocity is directly tied to how much charge crosses a section of the wire each second.
Visualizing Drift in a Wire
The arrows inside the wire suggest random electron motion, while the average drift of electrons is to the left. The conventional current is to the right.
Typical Size of Drift Velocity
One surprising fact is that drift velocity in ordinary wires is usually very small. Even when a current is noticeable, the electrons drift slowly, often on the order of millimeters per second or less.
This does not mean electric devices respond slowly. When a circuit is switched on, the electric field becomes established throughout the conductor very quickly, and charges everywhere in the wire begin drifting almost at once. The signal moves much faster than the individual electrons drift.
Example Calculation
Suppose a copper wire carries a current of $2.0\,\text{A}$. Let the wire have cross sectional area
$$
A = 1.0 \times 10^{-6}\,\text{m}^2
$$
and suppose the free electron number density is
$$
n = 8.5 \times 10^{28}\,\text{m}^{-3}
$$
Using $q = e = 1.60 \times 10^{-19}\,\text{C}$, the drift velocity is
$$
v_d = \frac{I}{nqA}
$$
$$
v_d = \frac{2.0}{(8.5 \times 10^{28})(1.60 \times 10^{-19})(1.0 \times 10^{-6})}
$$
$$
v_d \approx 1.47 \times 10^{-4}\,\text{m/s}
$$
So the drift velocity is about
$$
v_d \approx 0.15\,\text{mm/s}
$$
This is very slow compared with everyday speeds.
What Affects Drift Velocity
From the formula
$$
v_d = \frac{I}{nqA}
$$
we can see how drift velocity changes.
If current increases, drift velocity increases.
If the cross sectional area increases, drift velocity decreases, because more carriers can pass side by side.
If the number density $n$ increases, drift velocity decreases for the same current, because more charge carriers share the job of carrying the current.
The effect of charge magnitude $q$ is similar. Larger charge per carrier means less drift speed is needed to produce the same current.
Summary Table
| Quantity | Symbol | Meaning | SI unit |
|---|---|---|---|
| Current | $I$ | Rate of flow of charge | A |
| Drift velocity | $v_d$ | Average directed speed of charge carriers | m/s |
| Number density | $n$ | Carriers per unit volume | $\text{m}^{-3}$ |
| Charge per carrier | $q$ | Charge of one carrier | C |
| Cross sectional area | $A$ | Area of the wire cross section | $\text{m}^2$ |
Common Misunderstandings
A common misunderstanding is to think that electrons race through the wire at enormous speed because the lamp turns on immediately. In reality, the average drift velocity is small. What travels quickly is the electromagnetic influence through the circuit.
Another common misunderstanding is to confuse drift velocity with instantaneous velocity. An individual electron may move rapidly in random directions, but the drift velocity is the average net motion caused by the electric field.
Do not confuse these ideas:
Random thermal motion is large but averages to zero.
Drift velocity is small but gives a nonzero average motion.
Current depends on this average directed motion, not on the random motion alone.
Final Picture
Drift velocity provides the microscopic view of electric current. Current in a conductor is not produced because charges suddenly begin to exist, but because many charge carriers already present in the material acquire a small average drift under an electric field. This slow organized motion, summed over an enormous number of carriers, produces the current we measure in circuits.