Figure current i that is produced in a wire of resistivity 1.62 ×10^-8 Ω·m . The magnitude of th

step 1 For this problem on the topic of magnetism, we are shown a current i in the figure that is produ

Figure current i that is produced in a wire of resistivity...

Figure current $i$ that is produced in a wire of resistivity $1.62 \times 10^{-8} \Omega \cdot \mathrm{m} .$ The magnitude of the current versus time $t$ is shown in Fig. $32-35 b$. The vertical axisscale is set by $i_{s}=10.0 \mathrm{~A}$ and the horizontal axis scale is set by $t_{s}=50.0 \mathrm{~ms} .$ Point $P$ is at radial distance $9.00 \mathrm{~mm}$ from the wire's center. Determine the magnitude of the magnetic field $\vec{B}_{i}$ at point $P$ due to the actual current $i$ in the wire at (a) $t=20 \mathrm{~ms}$ (b) $t=40 \mathrm{~ms},$ and (c) $t=60 \mathrm{~ms}$ Next, assume that the electric field driving the current is confined to the wire. Then determine the magnitude of the magnetic field $\vec{B}_{i d}$ at point $P$ due to the displacement current $i_{d}$ in the wire at (d) $t=20 \mathrm{~ms},$ (e) $t=40 \mathrm{~ms},$ and (f) $t=60 \mathrm{~ms}$. At point $P$ at $t=20 \mathrm{~s}$ what is the direction (into or out of the page) of $(\mathrm{g}) \vec{B}_{i}$ and $(\mathrm{h}) \vec{B}_{i d} ?$

Figure current $i$ that is produced in a wire of resistivity $1.62 \times 10^{-8} \Omega \cdot \mathrm{m} .$ The magnitude of the current versus time $t$ is shown in Fig. $32-35 b$. The vertical axisscale is set by $i_{s}=10.0 \mathrm{~A}$ and the horizontal axis scale is set by $t_{s}=50.0 \mathrm{~ms} .$ Point $P$ is at radial distance $9.00 \mathrm{~mm}$ from the wire's center. Determine the magnitude of the magnetic field $\vec{B}_{i}$ at point $P$ due to the actual current $i$ in the wire at
(a) $t=20 \mathrm{~ms}$
(b) $t=40 \mathrm{~ms},$ and
(c) $t=60 \mathrm{~ms}$
Next, assume that the electric field driving the current is confined to the wire. Then determine the magnitude of the magnetic field $\vec{B}_{i d}$ at point $P$ due to the displacement current $i_{d}$ in the wire at
(d) $t=20 \mathrm{~ms},$ (e) $t=40 \mathrm{~ms},$ and (f) $t=60 \mathrm{~ms}$. At point $P$ at $t=20 \mathrm{~s}$ what is the direction (into or out of the page) of $(\mathrm{g}) \vec{B}_{i}$ and $(\mathrm{h}) \vec{B}_{i d} ?$

Key Concepts

Conduction Current

This is the flow of electric charge through a conducting material due to an applied electric field. In the context of electromagnetism, conduction currents generate magnetic fields that can be calculated using laws such as Ampere’s law or the Biot–Savart law. The analysis of conduction currents is fundamental in understanding how steady and time?dependent currents influence surrounding magnetic fields.

Displacement Current

Introduced by Maxwell, the displacement current is a term added to Ampere’s law to account for the effects of a time-varying electric field between regions where no real charge carriers are moving (i.e., in a capacitor gap). It ensures continuity of the magnetic field in scenarios where the electric field changes with time, playing an essential role in the full description of electromagnetic phenomena.

Ampere–Maxwell Law

This law extends Ampere’s law by including the displacement current so that it applies to both steady and time-varying fields. It is expressed as the curl of the magnetic field being proportional to the sum of the conduction current density and the rate of change of the electric field. This unified approach is central to Maxwell’s equations, which govern electromagnetism.

Magnetic Field Generation by Currents

The generation of a magnetic field from currents, whether conduction or displacement, is a core concept in electromagnetism. The magnitude and direction of the magnetic field depend on the current distribution and can be calculated using laws such as Ampere’s law or the Biot–Savart law. This principle is key to understanding how circuits and electromagnetic devices function.

Right-Hand Rule for Magnetic Fields

This is a mnemonic used to determine the direction of the magnetic field produced by an electric current. By orienting the fingers of the right hand in the direction of the conventional current, the thumb points in the direction of the resultant magnetic field. This rule is a fundamental tool for visualizing and predicting the behavior of magnetic fields in various electromagnetic scenarios.

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