A parallel plate capacitor (Fig. 8.7) made of circular...

A parallel plate capacitor (Fig. 8.7) made of circular plates each of radius $R=6.0 \mathrm{~cm}$ has a capacitance $C=100 \mathrm{pF}$. The capacitor is connected to a $230 \mathrm{~V}$ ac supply with a (angular) frequency of $300 \mathrm{rad} \mathrm{s}^{-1}$. (a) What is the rms value of the conduction current? (b) Is the conduction current equal to the displacement current? (c) Determine the amplitude of $\mathbf{B}$ at a point $3.0 \mathrm{~cm}$ from the axis between the plates.

A parallel plate capacitor (Fig. 8.7) made of circular plates each of radius $R=6.0 \mathrm{~cm}$ has a capacitance $C=100 \mathrm{pF}$. The capacitor is connected to a $230 \mathrm{~V}$ ac supply with a (angular) frequency of $300 \mathrm{rad} \mathrm{s}^{-1}$.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of $\mathbf{B}$ at a point $3.0 \mathrm{~cm}$ from the axis between the plates.

Key Concepts

Parallel Plate Capacitor

A parallel plate capacitor consists of two conductive plates separated by a dielectric. The capacitance, which measures the ability to store charge, depends on the area of the plates, the separation between them, and the properties of the dielectric medium. This device is fundamental in studying electrostatics, energy storage, and AC circuit behavior.

Capacitance

Capacitance is the property of a system that allows it to store electric charge per unit of potential difference. It is determined by the geometry of the conductors and the characteristics of the dielectric material between them. In many applications, capacitance is used to filter signals, store energy, or define time constants in circuits.

AC Circuits and RMS Current

In AC circuits, current and voltage vary sinusoidally with time. The root mean square (RMS) value of an AC quantity is a measure of its effective value equivalent to a DC value that would deliver the same power to a resistive load. RMS values are commonly used when analyzing power and energy in AC circuits.

Displacement Current

Displacement current is a concept introduced by Maxwell to extend Ampère's Law to situations where the electric field is time-varying, such as between the plates of a capacitor. Although not a flow of charges in the traditional sense, it is mathematically equivalent to a current and is essential to ensure continuity in the magnetic field equations in regions where no real current exists.

Ampère-Maxwell Law

The Ampère-Maxwell Law generalizes Ampère's Law by including the displacement current term. It relates the magnetic field circulating around a closed loop to both the conduction current and the time rate of change of the electric flux across a surface bounded by the loop. This law is central to understanding the generation of magnetic fields in time-varying electromagnetic environments.

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