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Ch 29: Electromagnetic Induction
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 29: Electromagnetic InductionProblem 42a
Chapter 29, Problem 42a

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 jD in the air space between the plates?

Verified step by step guidance
1
Understand the concept of displacement current: Displacement current is a term used in Maxwell's equations to account for the changing electric field in a capacitor. It is not a real current of moving charges, but it acts like a current in terms of creating a magnetic field.
Recall the formula for displacement current density: The displacement current density \( j_D \) is given by \( j_D = \frac{dE}{dt} \cdot \varepsilon_0 \), where \( \varepsilon_0 \) is the permittivity of free space and \( \frac{dE}{dt} \) is the rate of change of the electric field.
Relate the conduction current to the displacement current: In a charging capacitor, the conduction current \( I \) is equal to the displacement current \( I_D \). Therefore, \( I = I_D \).
Calculate the displacement current density: Use the relationship \( I_D = j_D \cdot A \), where \( A \) is the area of the capacitor plates. The area \( A \) can be calculated using \( A = \pi r^2 \), where \( r \) is the radius of the plates.
Substitute the values: Substitute the given values into the formula \( j_D = \frac{I}{A} \) to find the displacement current density. Use \( r = 4.00 \text{ cm} = 0.04 \text{ m} \) and \( I = 0.520 \text{ A} \).

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Key Concepts

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

Displacement Current

Displacement current is a concept introduced by Maxwell to account for the changing electric field in capacitors, which acts like a current in the space between the plates. It is given by the rate of change of electric flux and is crucial for understanding electromagnetic wave propagation and the continuity of current in circuits.
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Displacement vs. Distance

Capacitor Basics

A capacitor consists of two conductive plates separated by an insulating material, storing energy in the electric field created between the plates. The capacitance depends on the area of the plates, the distance between them, and the permittivity of the insulating material, influencing how the capacitor charges and discharges.
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Current Density

Current density is the amount of electric current flowing per unit area of a cross-section, typically measured in amperes per square meter. In the context of displacement current, it helps quantify the effect of a changing electric field in the space between capacitor plates, providing insight into the electromagnetic behavior of the system.
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Intro to Density
Related Practice
Textbook Question

The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. What is the emf between points a and b on the ring?

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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. (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?

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Textbook Question

The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. If the ring is cut at some point and the ends are separated slightly, what will be the emf between the ends?

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