Physics · Ch 11 — Electric Current Through Conductors

Drift Speed

11.4

Drift Speed

Picture a copper rod with no current flowing through it (Fig. 11.2): its free electrons are in constant random thermal motion, colliding repeatedly with the fixed ion cores of the lattice, but there is no net motion of the electron population as a whole in any particular direction -- the random velocities cancel out on average.

Now apply an electric field $\vec{E}$ along the length of the rod. Every free electron still moves randomly between collisions (this random component does not disappear), but superimposed on that randomness each electron now acquires a slow, systematic tendency to 'drift' in the direction OPPOSITE to $\vec{E}$ (since the electron's charge is negative, the electric force on it, $\vec{F}=-e\vec{E}$ , points opposite to the field). This organised (but very slow) net motion is called the drift speed, $V_d$ . In a typical copper conductor the drift speed is only of the order $10^{-4}$ to $10^{-5}$ m/s -- astonishingly slow compared with the electrons' own random thermal speed, which is of the order $10^{6}$ m/s. (The direction of an electric field at any point, incidentally, is simply defined as the direction of the force it would exert on a small positive test charge placed there.)

To connect this microscopic drift speed to the macroscopic current $I$ , consider a length $L$ of conducting wire of cross-sectional area $A$ (Fig. 11.3), and assume every free electron drifts with the same speed $V_d$ , with the current $I$ the same at every cross-section along the wire. If $n$ is the number of free electrons per unit volume, the total number of free electrons in this length $L$ of wire is $nAL$ , so the total (magnitude of) charge contained in this length is

q=nALe\qquad\text{--- (11.3)}

where $e$ is the magnitude of the electron's charge. This entire charge sweeps past any fixed cross-section of the wire in the time it takes an electron travelling at the drift speed to cover the length $L$ , namely

t=\frac{L}{V_d}\qquad\text{--- (11.4)}

Combining Eq. (11.1) with Eqs. (11.3) and (11.4), the current is

I=\frac{q}{t}=\frac{nALe}{L/V_d}=nAV_d e\qquad\text{--- (11.5)}

which can be rearranged to give the drift speed in terms of the current:

V_d=\frac{I}{nAe}=\frac{J}{ne}\qquad\text{--- (11.6)}

where $J=I/A$ is the CURRENT DENSITY -- the current per unit cross-sectional area, uniform over the cross-section $A$ of the wire, with SI unit A/m $^2$ :

$J=\frac{I}{A}\qquad\text{--- (11.7)}$ …

Figure 11.2Free electrons in random motion inside a conductor

What this figure shows. A schematic cross-section of a copper rod with NO current flowing through it (no external field applied). Many small dots representing free electrons are scattered throughout the interior of the rod, each drawn with a short zig-zag or multi-directional arrow trail indicating rapid, completely random thermal motion in every direction, with no preferred direction or net drift shown for the collection as a whole -- illustrating that in the absence of an applied field, the vector sum of all these random electron velocities average …

Figure 11.3Conducting wire with an applied electric field

What this figure shows. A section of a straight conducting wire of length L and uniform cross-sectional area A, drawn with an applied electric field vector E pointing along the length of the wire (say, left to right). Free electrons inside the wire are shown with a small drift-velocity arrow labelled Vd pointing in the direction OPPOSITE to E (right to left, since electrons are negative and are pushed against the field direction). The figure marks the cross-sectional area A at one end and the length L along the wire, setting up the derivation that the total charge crossing any cross-section in the time L/Vd (the time for an electron to traverse the marked …

Misc Ex.1Example 11.1 -- Drift velocity in a metallic wire from current and free-electron density

Worked out. A metallic wire of diameter 0.02 m contains 10^28 free electrons per cubic metre and carries a current of 100 A; using the cross-sectional area A = πr^2 (r = 0.01 m, so A ≈ 3.142×10^-4 m^2) and the relation Vd = I/(nAe) with e = 1.6×10^-19 C, the worked solution substitutes the numbers directly to obtain the drift velocity, illustrating just how slow drift motion is (of order 10^-4 m/s) compared with the much faster random thermal speed of the same electrons (order 10^6 m/s). …

Misc Ex.2Example 11.2 -- Current density in a copper wire from current and radius

Worked out. A copper wire of radius 0.6 mm carries a current of 1 A, assumed uniformly distributed over its cross-section; the worked solution computes the cross-sectional area A = πr^2 (r = 0.6×10^-3 m, giving A ≈ 1.1311×10^-6 m^2) and then applies J = I/A directly to obtain the current density in A/m^2, a straightforward numeric application of the current-density definition introduced in this section. …

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