27.3.1 The Current Density in a Wire

We found the electron current in a wire of cross-section area $A$ to be $i_{e} = n_{e} A v_{d}$ . Thus the current $I$ is

I = e i_{e} = n_{e} e v_{d} A

(27.12)

The quantity $n_{e} e v_{d}$ depends on the charge carriers and on the internal electric field that determines the drift speed, whereas $A$ is simply a physical dimension of the wire. It will be useful to separate these quantities by defining the current density $J$ in a wire as the current per square meter of cross section:

J = current density \equiv \frac{I}{A} = n_{e} e v_{d}

(27.13)

The current density has units of ${A/m}^{2}$ . A specific piece of metal, shaped into a wire with cross-section area $A$ , carries current $I = J A$ .

EXAMPLE 27.4 Finding the electron drift speed

A 1.0 A current passes through a 1.0-mm-diameter copper wire. What are the current density and the drift speed of the electrons in the wire?

SOLVE We can find the drift speed from the current density. The current density is

J = \frac{I}{A} = \frac{I}{π r^{2}} = \frac{1.0 A}{π (0.00050 {m)}^{2}} = 1.3 \times 10^{6} {A/m}^{2}

The electron drift speed is thus

v_{d} = \frac{J}{n_{e} e} = 9.6 \times 10^{- 5} m / s = 0.096 mm / s

where the conduction-electron density for copper was taken from Table 27.1.

REVIEW We earlier used $1.0 \times 10^{- 4} m / s$ as a typical electron drift speed. This example shows where that value comes from.