An aluminum wire with a diameter of 0.100 mm has a uniform...

An aluminum wire with a diameter of $0.100 \mathrm{~mm}$ has a uniform electric field with a magnitude of $0.200 \mathrm{~V} / \mathrm{m}$ imposed along its entire length. The temperature of the wire is $50.0^{\circ} \mathrm{C}$. Assume one free electron per atom. (a) Using the information given in Table $27.1$, determine the resistivity. (b) What is the current density in the wire? (c) What is the total current in the wire? (d) What is the drift speed of the conduction electrons? (e) What potential difference must exist between the ends of a $2.00-\mathrm{m}$ length of the wire if the stated electric field is to be produced?

An aluminum wire with a diameter of $0.100 \mathrm{~mm}$ has a uniform electric field with a magnitude of $0.200 \mathrm{~V} / \mathrm{m}$ imposed along its entire length. The temperature of the wire is $50.0^{\circ} \mathrm{C}$. Assume one free electron per atom.
(a) Using the information given in Table $27.1$, determine the resistivity. (b) What is the current density in the wire? (c) What is the total current in the wire?
(d) What is the drift speed of the conduction electrons?
(e) What potential difference must exist between the ends of a $2.00-\mathrm{m}$ length of the wire if the stated electric field is to be produced?

Key Concepts

Free Electron Concentration

Free electron concentration is the number of charge carriers per unit volume available to conduct electricity in a material. In metals, this value is typically high, and assuming a specific number of free electrons per atom allows for the calculation of current flow characteristics such as drift velocity and current density.

Drift Velocity

Drift velocity refers to the average velocity at which free electrons move through a conductor under the influence of an electric field. It is a key microscopic parameter that connects to the macroscopic current by indicating how the random motion of electrons results in a net movement in one direction when an external field is applied.

Resistivity

Resistivity is an intrinsic property of materials that quantifies how strongly a given material opposes the flow of electric current. It is temperature dependent and critical in determining the ease with which current flows through a conductor. The lower the resistivity, the better the material conducts electricity, and it is used alongside the conductor's dimensions in calculating resistance.

Electric Field and Potential Difference

The electric field is the force per unit charge that drives the flow of electrons through a conductor. It is related to the potential difference (voltage) across a material and its length, as given by the relationship E = V/L. Understanding how an applied electric field creates a potential difference is crucial for analyzing current flow in circuits.

Current Density

Current density is defined as the electric current per unit cross-sectional area of a conductor. It provides a measure of how concentrated the current is within a material and is calculated by dividing the total current by the cross-sectional area. This concept is important for linking macroscopic conductivity with microscopic charge movement.

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