Copper has 8.5 ×10^28 free electrons per cubic meter. A...

Copper has $8.5 \times 10^{28}$ free electrons per cubic meter. A 71.0-cm length of 12-gauge copper wire that is 2.05 mm in diameter carries 4.85 A of current. (a) How much time does it take for an electron to travel the length of the wire? (b) Repeat part (a) for 6-gauge copper wire (diameter 4.12 mm) of the same length that carries the same current. (c) Generally speaking, how does changing the diameter of a wire that carries a given amount of current affect the drift velocity of the electrons in the wire?

Key Concepts

Electron Drift Velocity

Electron drift velocity is the average velocity at which free electrons in a conductor move when an electric field is applied. Although electrons move randomly at high speeds due to thermal energy, their net movement in one direction is relatively slow. This concept is quantitatively described by the relationship v_d = I / (n q A), where I is the current, n is the free electron density, q is the elementary charge, and A is the cross-sectional area of the conductor.

Free Electron Density

Free electron density refers to the number of electrons available per unit volume in a material that can move under the influence of an electric field. In conductive metals, a high free electron density means more charge carriers are available, which impacts the material's electrical conductivity and the resulting drift velocity for a given current.

Wire Cross-Sectional Area

The cross-sectional area of a wire, determined by its diameter or geometry, is crucial in current conduction as it affects the number of charge carriers available across the wire's cross section. For a constant current, a larger cross-sectional area leads to lower current density and slower drift velocity of the electrons, while a smaller area results in a higher drift velocity.

Relationship Between Current, Charge Density, and Drift Velocity

This relationship is fundamental to understanding electron movement in conductors. It is encapsulated by the equation I = n q A v_d, which links the current flowing through the conductor to the free electron density, the elementary charge, the cross-sectional area of the wire, and the electron drift velocity. This relation helps predict how changes in physical dimensions or material properties influence the behavior of charge carriers in a circuit.

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