A copper wire joins an aluminum wire whose diameter is twice...

A copper wire joins an aluminum wire whose diameter is twice that of the copper. The same current flows in both wires. The density of conduction electrons in copper is $1.1 \times 10^{29} \mathrm{~m}^{-3}$; in aluminum it's $2.1 \times 10^{29} \mathrm{~m}^{-3}$. Compare (a) the drift speeds and (b) the current densities in each wire.

Key Concepts

Conduction Electron Density

The conduction electron density is the number of free electrons per unit volume available to conduct electric current in a material. It directly affects the drift velocity for a given current: higher electron density results in lower required drift velocities. This parameter is fundamental in comparing conduction behavior across different materials.

Influence of Cross-sectional Area

The cross-sectional area of a conductor is crucial in determining both current density and drift velocity. For wires with different diameters, differences in area change the distribution of current and must be accounted for when comparing microscopic conduction factors. The area influences the current density, which in turn affects the drift velocity needed to sustain the same overall current.

Drift Velocity

Drift velocity is the average velocity at which conduction electrons move through a conductor under an applied electric field. It is calculated via the relation I = n q A v_d, where I is the current, n is the conduction electron density, q is the elementary charge, A is the cross-sectional area, and v_d is the drift velocity. This concept links the microscopic behavior of electrons to the macroscopic observable of current.

Current Density

Current density is defined as the amount of electrical current flowing per unit area of a conductor, given by j = I/A. It connects the macroscopic current with the geometry of the conductor, allowing analysis of how changes in cross-sectional area affect the conduction process. When combined with the drift velocity relation, it provides insight into how different factors such as conduction electron density influence the conduction properties.

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