A length of metal wire has a radius of 5.00 ×10^-3 m and a resistance of 0.100Ω. When the potent

step 1 Let's say we have a wire with a cross -sectional area. A and the length L, for this wire, we can

A length of metal wire has a radius of 5.00 ×10^-3 m and a...

A length of metal wire has a radius of $5.00 \times 10^{-3} \mathrm{m}$ and a resistance of 0.100$\Omega .$ When the potential difference across the wire is 15.0 $\mathrm{V}$ , the electron drift speed is found to be $3.17 \times$ $10^{-4} \mathrm{m} / \mathrm{s}$ , On the basis of these data, calculate the density of free electrons in the wire.

Key Concepts

Free Electron Density

Free electron density (or charge carrier density) refers to the number of conduction electrons per unit volume in a material. This key parameter is critical in determining the electrical properties of conductors. A higher free electron density typically means that the material has more carriers available to conduct electricity, directly influencing quantities such as conductivity and the observed current under the application of a potential difference.

Ohm’s Law and Resistivity

Ohm's Law establishes a linear relationship between the voltage across a conductor, the current flowing through it, and its resistance. This fundamental law, coupled with the concept of resistivity—a material-specific property that quantifies how strongly a material opposes the flow of electric current—allows us to connect macroscopic electrical measurements with the underlying microscopic charge dynamics. Through the geometry of the conductor and the measured resistance, one can derive related parameters such as current density and, subsequently, free electron density.

Drift Velocity

Drift velocity is the average velocity that charged particles, such as electrons, attain due to an electric field applied across a conductor. Even though the random thermal motion of electrons is very high, their net movement in response to an electric field is typically very small. This concept is central to understanding how a current is established in a conductor and allows us to relate measurable quantities like current and applied voltage to the microscopic behavior of charge carriers.

Current Density

Current density is defined as the electric current per unit cross-sectional area flowing through a conductor. It serves as a bridge between the macroscopic current observable in a circuit and the microscopic motion of electrons. By relating current density to the electron drift speed and the charge density, one can gain insight into the number of carriers moving within the conductor and their collective contribution to the overall current.

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