A silver wire has resistivity ρ=1.62 ×10^-3 R ·m and a crosssectional area of 5.00 mm^2. The cu

step 1 Hello everyone. Part A from Omslaw we have E is equal to row into J and J is equal to I by A so

A silver wire has resistivity ρ=1.62 ×10^-3 R ·m and a...

A silver wire has resistivity $\rho=1.62 \times 10^{-3} \mathrm{R} \cdot \mathrm{m}$ and a crosssectional area of $5.00 \mathrm{~mm}^{2}$. The current in the wire is uniform and changing at the rate of 2000 A/s when the current is $54.0 \mathrm{~A}$. (a) What is the magnitude of the (uniform) electric field in the wire when the current in the wire is $100 \mathrm{~A}$ ? (b) What is the displacement current in the wire at that time? (c) What is the ratio of the magnitude of the magnetic field due to the displacement current to that due to the current at a distance $r$ from the wire?

Key Concepts

Resistivity

Resistivity is a material-specific property that quantifies how strongly a material opposes the flow of electric current. It appears in Ohm’s law in its microscopic form, relating the electric field in a material to the current density. A lower resistivity indicates that the material conducts electricity more easily.

Electric Field in a Conductor

The electric field in a conductor is directly related to the current density and the material’s resistivity. It is defined by the relation E = ?J, where ? is the resistivity and J is the current density. This concept is essential to understand how voltage drives current through a material and determines the forces experienced by charge carriers.

Current Density

Current density is a measure of the electric current per unit cross-sectional area of the conductor. It is defined by the equation J = I/A, where I is the current and A is the cross-sectional area. This parameter is crucial for linking macroscopic current to the microscopic electric field within a material.

Displacement Current

Displacement current is a term added by Maxwell to Ampere’s law to account for the changing electric field in regions where there is no conduction current, ensuring consistency with the conservation of charge. It is given by the rate of change of the electric displacement field, and although it does not involve the movement of actual charges, it produces magnetic fields similar to conduction currents.

Maxwell’s Correction to Ampere’s Law

Maxwell’s correction to Ampere’s law involves adding the displacement current term to the conduction current. This corrected form allows for a complete description of the generation of magnetic fields in scenarios where the electric field is time-dependent, thereby unifying the treatment of static and dynamic electromagnetic fields.

Magnetic Field due to Conduction and Displacement Currents

The magnetic field due to a current in a conductor can be calculated using Ampere’s law, which considers the conduction current. When the current changes with time, a displacement current is induced, producing its own magnetic field. Comparing these fields, often through their ratio, helps illustrate the relative contributions of conduction and displacement effects under dynamic conditions.

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