Ch 29: Electromagnetic Induction

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 29: Electromagnetic Induction

Problem 42cd

Chapter 29, Problem 42cd

A parallel-plate, air-filled capacitor is being charged as in Fig. 29.23. The circular plates have radius 4.00 cm, and at a particular instant the conduction current in the wires is 0.520 A. (c) What is the induced magnetic field between the plates at a distance of 2.00 cm from the axis? (d) At 1.00 cm from the axis?

Verified step by step guidance

Understand that the problem involves calculating the induced magnetic field between the plates of a parallel-plate capacitor. This is related to the displacement current, which is a concept introduced by Maxwell to account for the changing electric field in capacitors.

Use Ampère-Maxwell Law, which is an extension of Ampère's Law, to include the displacement current. The law is given by:

\int B \cdot d s = \frac{μ o}{I}

_c+μoεo∮dΦ_Edt, where

I

_c is the conduction current and

d Φ

_Edt is the rate of change of electric flux.

Calculate the displacement current

I

_d using the formula:

I

_d=εodΦ_Edt. Since the conduction current

I

_c is given as 0.520 A, the displacement current is equal to the conduction current in this scenario.

Apply the Ampère-Maxwell Law to find the magnetic field

B

at a distance

r

from the axis. For a distance of 2.00 cm, use the formula:

B = μ o I

_d2πr, where

μ o

is the permeability of free space.

Repeat the calculation for a distance of 1.00 cm from the axis using the same formula:

B = μ o I

_d2πr. This will give you the magnetic field at the specified distance from the axis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ampere-Maxwell Law

The Ampere-Maxwell Law extends Ampere's Law by incorporating the displacement current, allowing for the calculation of magnetic fields in situations where electric fields change over time. It is crucial for understanding the magnetic field between capacitor plates as it accounts for both conduction and displacement currents.

Displacement Current

Displacement current is a concept introduced by Maxwell to account for changing electric fields in capacitors, which produce magnetic fields similar to conduction currents. It is essential for analyzing the magnetic field between the plates of a charging capacitor, as it complements the conduction current in the wires.

Magnetic Field in a Capacitor

The magnetic field between the plates of a capacitor can be determined using the modified Ampere's Law, considering both conduction and displacement currents. This field varies with distance from the axis, and understanding its calculation is key to solving the problem of induced magnetic fields at specific radial distances.

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Related Practice

Textbook Question

A parallel-plate, air-filled capacitor is being charged as in Fig. 29.23. The circular plates have radius 4.00 cm, and at a particular instant the conduction current in the wires is 0.520 A. What is the rate at which the electric field between the plates is changing?

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Textbook Question

A long, straight solenoid with a cross-sectional area of 8.00 cm² is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

Textbook Question

A parallel-plate, air-filled capacitor is being charged as in Fig. 29.23. The circular plates have radius 4.00 cm, and at a particular instant the conduction current in the wires is 0.520 A. (a) What is the displacement current density j_D in the air space between the plates?

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