11.9 Drift Velocity

Drift velocity is the average velocity attained by charge carriers, such as electrons, in a material due to an electric field. While electrons in a conductor are in constant, high-speed random motion, the application of an electric field superimposes a slow, directional drift on this random movement, resulting in a net flow of charge, which is called electric current.

Random Motion vs. Drift Motion

• Without an Electric Field: In a conductor, free electrons move randomly at very high speeds (the Fermi velocity, on the order of $10^6$ m/s). Their motion is chaotic, and they frequently collide with the atoms of the conductor's lattice. Because the motion is random, there is no net movement of charge in any particular direction, so there is no current.

• With an Electric Field: When an external electric field is applied (e.g., by connecting a battery), a force is exerted on the electrons, causing them to accelerate in the direction opposite to the field. However, their path is constantly interrupted by collisions with the lattice. The result is not a continuous acceleration but a slow, average drift velocity in a specific direction.

Magnitude of Drift Velocity

The drift velocity of electrons is surprisingly slow, typically on the order of millimeters per second ($10^{-4}$ to $10^{-5}$ m/s). This is in stark contrast to their high-speed random motion.

Relationship with Electric Current

Drift velocity is directly responsible for electric current. The current ($I$) flowing through a conductor can be expressed by the formula:

$I=nA{v}_{d}e$

Where:

• $I$ = Electric current (in Amperes)
• $n$ = Number of free charge carriers per unit volume (charge carrier density)
• $A$ = Cross-sectional area of the conductor
• $v_d$ = Drift velocity of the charge carriers
• $e$ = Charge of a single carrier (for electrons, $e\approx 1.6\times 10^{-19}$ C)

This equation shows that for a given wire, the current is directly proportional to the drift velocity.

Mobility ($\mu$)

Mobility is a property of a charge carrier that measures how quickly it can move through a material in response to an electric field. It is the ratio of the drift velocity to the electric field strength ($E$).

• Formula: $\mu =\frac{v_d}{E}$
• Unit: m²/(V·s)

Example Calculation

Problem: A copper wire has a diameter of 2.05 mm and carries a current of 10 A. Copper has $8.5\times 10^{28}$ free electrons per cubic meter. What is the drift velocity of the electrons?

Solution:

1. Find the cross-sectional area (A):

   • Radius $r=\frac{2.05\text{ mm}}{2}=1.025\times 10^{-3}\text{ m}$
   • Area $A=\pi r^2=\pi (1.025\times 10^{-3}{)}^2\approx 3.3\times 10^{-6}{\text{ m}}^2$

2. Use the current formula to solve for $v_d$: $v_d=\frac{I}{nAe}$

3. Substitute the values:

   • $I=10\text{ A}$
   • $n=8.5\times 10^{28}{\text{ m}}^{-3}$
   • $A\approx 3.3\times 10^{-6}{\text{ m}}^2$
   • $e=1.6\times 10^{-19}\text{ C}$ $v_d=\frac{10}{(8.5\times 10^{28})(3.3\times 10^{-6})(1.6\times 10^{-19})}$ $v_d\approx 2.23\times 10^{-4}\text{m/s}\text{(or 0.223 mm/s)}$

The drift velocity is extremely slow, at only about 0.223 millimeters per second.

Possible Questions and Answers

Q: If drift velocity is so slow, why does a light bulb turn on almost instantly when you flip the switch?

A: This is a common misconception. While individual electrons drift slowly, the electric field that drives them propagates through the wire at nearly the speed of light. When you flip the switch, this field is established almost instantaneously throughout the entire circuit. It is this field that causes all the free electrons along the wire to start drifting at the same time, including the ones already in the light bulb filament. It is like a long tube filled with marbles; when you push one in at one end, one immediately comes out the other end, even though no single marble traveled the full length.