Calculating Drift Velocity in a Common Wire

We can calculate the drift velocity using the equation $I={\mathrm{nqAv}}_{\text{d}}$. The current $I=\mathrm{20.0\; A}$ is given, and $q=–1.60\times {10}^{–19}\text{C}$ is the charge of an electron. We can calculate the area of a cross-section of the wire using the formula $A=\pi r^2,$ where $r$ is one-half the given diameter, 2.053 mm. We are given the density of copper, $8.80\times {10}^3{\text{kg/m}}^3,$ and the periodic table shows that the atomic mass of copper is 63.54 g/mol. We can use these two quantities along with Avogadro’s number, $6.02\times {10}^{23}\text{atoms/mol},$ to determine $n,$ the number of free electrons per cubic meter.

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

The cross-sectional area of the wire is

Rearranging $I=nqA{v}_{\text{d}}$ to isolate drift velocity gives

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 ${\text{10}}^{-4}\text{m/s}$) confirms that the signal moves on the order of ${\text{10}}^{\text{12}}$ times faster (about ${\text{10}}^8\text{m/s}$) than the charges that carry it.