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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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.