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Picture is a schematic drawing of charges q flowing from left to right with the speed Vd through the wire with the cross-sectional area A.
All the charges in the shaded volume of this wire move out in a time dt , having a drift velocity of magnitude v d .

Note that simple drift velocity is not the entire story. The speed of an electron is sometimes much greater than its drift velocity. In addition, not all of the electrons in a conductor can move freely, and those that do move might move somewhat faster or slower than the drift velocity. So what do we mean by free electrons?

Atoms in a metallic conductor are packed in the form of a lattice structure. Some electrons are far enough away from the atomic nuclei that they do not experience the attraction of the nuclei as strongly as the inner electrons do. These are the free electrons. They are not bound to a single atom but can instead move freely among the atoms in a “sea” of electrons. When an electrical field is applied, these free electrons respond by accelerating. As they move, they collide with the atoms in the lattice and with other electrons, generating thermal energy, and the conductor gets warmer. In an insulator, the organization of the atoms and the structure do not allow for such free electrons.

As you know, electric power is usually supplied to equipment and appliances through round wires made of a conducting material (copper, aluminum, silver, or gold) that are stranded or solid. The diameter of the wire determines the current-carrying capacity—the larger the diameter, the greater the current-carrying capacity. Even though the current-carrying capacity is determined by the diameter, wire is not normally characterized by the diameter directly. Instead, wire is commonly sold in a unit known as “gauge.” Wires are manufactured by passing the material through circular forms called “drawing dies.” In order to make thinner wires, manufacturers draw the wires through multiple dies of successively thinner diameter. Historically, the gauge of the wire was related to the number of drawing processes required to manufacture the wire. For this reason, the larger the gauge, the smaller the diameter. In the United States, the American Wire Gauge (AWG) was developed to standardize the system. Household wiring commonly consists of 10-gauge (2.588-mm diameter) to 14-gauge (1.628-mm diameter) wire. A device used to measure the gauge of wire is shown in [link] .

Picture is a photograph of a device used for measuring the gauge of electrical wire. The higher gauge numbers indicate thinner wires.
A device for measuring the gauge of electrical wire. As you can see, higher gauge numbers indicate thinner wires.

Calculating drift velocity in a common wire

Calculate the drift velocity of electrons in a copper wire with a diameter of 2.053 mm (12-gauge) carrying a 20.0-A current, given that there is one free electron per copper atom. (Household wiring often contains 12-gauge copper wire, and the maximum current allowed in such wire is usually 20.0 A.) The density of copper is 8.80 × 10 3 kg/m 3 and the atomic mass of copper is 63.54 g/mol.

Strategy

We can calculate the drift velocity using the equation I = n q A v d . The current is I = 20.00 A and q = 1.60 × 10 −19 C is the charge of an electron. We can calculate the area of a cross-section of the wire using the formula A = π r 2 , where r is one-half the diameter. The given diameter is 2.053 mm, so r is 1.0265 mm. We are given the density of copper, 8.80 × 10 3 kg/m 3 , and the atomic mass of copper is 63.54 g/mol . We can use these two quantities along with Avogadro’s number, 6.02 × 10 23 atoms/mol , to determine n , the number of free electrons per cubic meter.

Solution

First, we calculate the density of free electrons in copper. There is one free electron per copper atom. Therefore, the number of free electrons is the same as the number of copper atoms per m 3 . We can now find n as follows:

n = 1 e atom × 6.02 × 10 23 atoms mol × 1 mol 63.54 g × 1000 g kg × 8.80 × 10 3 kg 1 m 3 = 8.34 × 10 28 e / m 3 .

The cross-sectional area of the wire is

A = π r 2 = π ( 2.05 × 10 −3 m 2 ) 2 = 3.30 × 10 −6 m 2 .

Rearranging I = n q A v d to isolate drift velocity gives

v d = I n q A = 20.00 A ( 8.34 × 10 28 / m 3 ) ( −1.60 × 10 −19 C)( 3.30 × 10 −6 m 2 ) = −4.54 × 10 −4 m/s .

Significance

The minus sign indicates that the negative charges are moving in the direction opposite to conventional current. The small value for drift velocity (on the order of 10 −4 m/s ) confirms that the signal moves on the order of 10 12 times faster (about 10 8 m/s ) than the charges that carry it.

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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