Electric Current

Electric current is defined as the rate at which electric charge flows. A large current—such as the current used to start a truck engine—moves a large amount of charge in a short time. A small current—such as the one used to operate a handheld calculator—moves a smaller amount of charge over a longer time. In equation form, electric current
[latex]I[/latex] is defined by:

Here, [latex]\Delta Q[/latex] is the amount of charge that passes through a given cross-sectional area during the time interval [latex]\Delta t[/latex]. (Often the initial time is taken as zero, so [latex]\Delta t=t[/latex].) See Figure 19.1.
The SI unit for current is the ampere (A), named for the French physicist André-Marie Ampère (1775–1836). Since
[latex]I=\Delta Q/\Delta t[/latex], an ampere is one coulomb per second:

Current ratings appear everywhere in health and everyday technology—for example, on fuses and circuit breakers (patient-room outlets and medical devices must be protected from overload), as well as on battery packs, chargers, and many electronic instruments.

Figure 19.2 shows a simple circuit and its schematic representation: a battery, conducting path, and a load (shown as a resistor). Schematics help us focus on the essential features of a circuit. The same schematic in Figure 19.2(b) could represent many systems—from a truck headlight to a small penlight—because the physics of current and energy transfer is analyzed in the same way.

Notice that the current direction shown in Figure 19.2 is from positive to negative. By definition,
conventional current points in the direction that positive charge would flow.
Depending on the material, the moving charges may be positive, negative, or both. In metal wires, current is carried mainly by electrons (negative charges). In ionic solutions—such as salt water or the fluid inside and outside cells—both positive and negative ions can move, and this is directly relevant to nerve and muscle signaling.

It is also important to recognize that an electric field exists inside a conductor carrying current (see Figure 19.3). Unlike a conductor in electrostatic equilibrium (where the internal electric field is zero), a conductor with current is not in static equilibrium. The electric field provides the push that transfers energy to the charge carriers.

Drift Velocity

Electrical signals appear to move extremely quickly. A phone call carried by electrical signals can travel long distances with no noticeable delay, and a light turns on as soon as a switch is flipped. In many circuits, the signal (a change in electric field and charge distribution) propagates at speeds on the order of
[latex]10^{8}~\text{m/s}[/latex], a significant fraction of the speed of light.
Surprisingly, the individual charge carriers that make up the current move much more slowly on average, typically with drift speeds around
[latex]10^{-4}~\text{m/s}[/latex].

The rapid signal speed occurs because electric forces act quickly across the conductor. When a charge is pushed into a wire, it repels nearby charges, which repel charges farther down the line. Because charge density in a conductor cannot increase easily, an incoming charge effectively “forces” another charge to leave the far end almost immediately (see Figure 19.4). More precisely, the quickly moving “shock wave” is a rapidly propagating change in the electric field throughout the conductor.

Good conductors contain large numbers of mobile charge carriers. In metals, these carriers are free electrons. Figure 19.5 shows that individual electrons undergo many collisions with atoms and other electrons, so their motion is largely random—similar to the random motion of gas molecules. However, an electric field exists inside the wire, and it creates a small net motion superimposed on this random motion. The drift velocity [latex]v_{\text{d}}[/latex] is the average velocity of the charge carriers (for electrons, the drift is opposite the electric field).

Collisions of free electrons with the lattice atoms transfer energy to the material. The electric field does work on the charges, but in a steady current that work does not build up as increasing electron speed; instead, the work is transferred to the conductor as thermal energy, often raising its temperature. That is why a continual power input is required to maintain current in ordinary conductors.
An important exception is a superconductor, which can sustain current with essentially no energy loss.

We can relate drift velocity to current by counting the charge carriers in a segment of wire, as shown in Figure 19.6. Let [latex]n[/latex] be the number of free charges per unit volume (the carrier density). A shaded segment of wire with cross-sectional area [latex]A[/latex] and length [latex]x[/latex] has volume [latex]Ax[/latex], so it contains [latex]nAx[/latex] carriers. If each carrier has charge [latex]q[/latex], then the charge in that segment is [latex]\Delta Q = qnAx[/latex]. If all of that charge moves out of the segment in time [latex]\Delta t[/latex], then:

Since the carriers move an average distance [latex]x[/latex] in time [latex]\Delta t[/latex], the magnitude of the drift velocity is
[latex]v_{\text{d}} = x/\Delta t[/latex]. Substituting [latex]x/\Delta t=v_{\text{d}}[/latex] gives:

In this expression, [latex]I[/latex] is the current through a wire of cross-sectional area [latex]A[/latex]. The material has carrier density [latex]n[/latex], each carrier has charge [latex]q[/latex], and the carriers drift with average speed [latex]v_{\text{d}}[/latex]. This relationship is useful in both directions: given a current, it can predict drift velocity; given drift velocity, it can predict current.

Drift velocity is only part of the story. Individual electrons move much faster than [latex]v_{\text{d}}[/latex] due to their random thermal motion; the drift velocity is the small net average caused by the electric field. Also, not every electron in a metal is free to move throughout the material. In a metal lattice, some electrons are loosely bound and can move through a “sea” of electrons shared by the atoms. These are the free electrons that respond to an applied electric field, collide with the lattice, and transfer energy to the material as heat. In an insulator, electrons are more tightly bound and cannot move freely, so current is much smaller.