Six copper wires are characterized by their dimensions and...

Six copper wires are characterized by their dimensions and by the current they carry. Rank the wires in order of decreasing drift velocity. (a) diameter $2 \mathrm{mm}$, length $2 \mathrm{m}$, current $80 \mathrm{mA}$ (b) diameter $1 \mathrm{mm}$, length $1 \mathrm{m}$, current $80 \mathrm{mA}$ (c) diameter $4 \mathrm{mm}$, length $16 \mathrm{m}$, current $40 \mathrm{mA}$ (d) diameter $2 \mathrm{mm}$, length $2 \mathrm{m}$, current $160 \mathrm{mA}$ (e) diameter $1 \mathrm{mm}$, length $4 \mathrm{m}$, current $20 \mathrm{mA}$ (f) diameter $2 \mathrm{mm}$, length $1 \mathrm{m}$, current $40 \mathrm{mA}$

Six copper wires are characterized by their dimensions and by the current they carry. Rank the wires in order of decreasing drift velocity.
(a) diameter $2 \mathrm{mm}$, length $2 \mathrm{m}$, current $80 \mathrm{mA}$
(b) diameter $1 \mathrm{mm}$, length $1 \mathrm{m}$, current $80 \mathrm{mA}$
(c) diameter $4 \mathrm{mm}$, length $16 \mathrm{m}$, current $40 \mathrm{mA}$
(d) diameter $2 \mathrm{mm}$, length $2 \mathrm{m}$, current $160 \mathrm{mA}$
(e) diameter $1 \mathrm{mm}$, length $4 \mathrm{m}$, current $20 \mathrm{mA}$
(f) diameter $2 \mathrm{mm}$, length $1 \mathrm{m}$, current $40 \mathrm{mA}$

Key Concepts

Cross-Sectional Area

The cross-sectional area of a conductor is a critical geometric parameter that affects the drift velocity. It is calculated based on the wire’s dimensions (typically using the diameter for circular cross-sections). Since drift velocity is inversely proportional to the cross-sectional area, a smaller area results in a larger current density and hence a higher drift velocity.

Independence from Wire Length

In the formula for drift velocity, the length of the conductor does not appear, highlighting that drift velocity is independent of the wire's length. Length might affect resistance and the overall voltage drop across a conductor, but it does not directly impact the drift velocity, which depends solely on current, charge carrier density, and cross-sectional area.

Drift Velocity

Drift velocity is the average velocity that electrons (or charge carriers) acquire in a conductor when subjected to an electric field. It is mathematically given by the relation v_d = I/(nAe), where I is the current, n is the charge carrier density, A is the cross-sectional area, and e is the elementary charge. This concept is fundamental in understanding how current flows in materials.

Current Density

Current density is defined as the current per unit cross-sectional area of a conductor, expressed as j = I/A. It directly influences drift velocity, since a higher current density (resulting from a high current or a smaller cross-sectional area) leads to a higher drift velocity, all else being equal.

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