(III) (a) When a circular parallel-plate capacitor is being charged as in Example 31–1, show that the Poynting vector S→\overrightarrow{S} points radially inward toward the center of the capacitor, parallel to the plates.

Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Problem 34a

Chapter 30, Problem 34a

(III) (a) When a circular parallel-plate capacitor is being charged as in Example 31–1, show that the Poynting vector $\vec{S}$ points radially inward toward the center of the capacitor, parallel to the plates.

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Understand the Poynting vector: The Poynting vector $ \mathbf{S} $ represents the flow of electromagnetic energy and is given by $ \mathbf{S} = \mathbf{E} \times \mathbf{B} / \mu_0 $, where $ \mathbf{E} $ is the electric field, $ \mathbf{B} $ is the magnetic field, and $ \mu_0 $ is the permeability of free space.

Analyze the electric field $ \mathbf{E} $: In a parallel-plate capacitor, the electric field $ \mathbf{E} $ is uniform and directed perpendicular to the plates, pointing from the positively charged plate to the negatively charged plate.

Examine the magnetic field $ \mathbf{B} $: During charging, the current flowing into the capacitor creates a magnetic field $ \mathbf{B} $ that encircles the current path. Using the right-hand rule, $ \mathbf{B} $ forms concentric circles around the axis of the capacitor plates.

Determine the direction of $ \mathbf{S} $: The Poynting vector $ \mathbf{S} $ is the cross product of $ \mathbf{E} $ and $ \mathbf{B} $. Since $ \mathbf{E} $ is perpendicular to the plates and $ \mathbf{B} $ is tangential to the circular path, $ \mathbf{S} $ points radially inward toward the center of the capacitor.

Conclude the energy flow: The inward direction of $ \mathbf{S} $ indicates that electromagnetic energy flows into the capacitor, parallel to the plates, as it is being charged.

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Key Concepts

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

Poynting Vector

The Poynting vector, denoted as →S, represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is calculated using the cross product of the electric field vector (→E) and the magnetic field vector (→B), expressed as →S = (1/μ₀) →E × →B. In the context of a capacitor, the Poynting vector indicates the flow of electromagnetic energy, which is crucial for understanding how energy is transferred during the charging process.

Electric Field in Capacitors

In a parallel-plate capacitor, an electric field (→E) is established between the plates when a voltage is applied. This field is uniform and directed from the positive plate to the negative plate, with a strength proportional to the voltage and inversely proportional to the distance between the plates. Understanding the behavior of the electric field is essential for analyzing how it interacts with the magnetic field during the charging process.

Magnetic Field and Charging

When a capacitor is charged, a changing electric field generates a magnetic field (→B) around the capacitor, as described by Maxwell's equations. This magnetic field is oriented in a direction that is perpendicular to the electric field and the direction of energy flow. The relationship between the electric and magnetic fields is fundamental to understanding the dynamics of electromagnetic waves and the behavior of the Poynting vector in the context of the charging capacitor.

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Textbook Question

(a) When a circular parallel-plate capacitor is being charged as in Example 31–1, show that the Poynting vector $\vec{S}$ points radially inward toward the center of the capacitor, parallel to the plates.

(b) Integrate $\vec{S}$ over the cylindrical boundary of the capacitor gap to show that the rate at which energy enters the capacitor is equal to the rate at which electrostatic energy is being stored in the electric field of the capacitor (Section 24–4). Ignore fringing of $\vec{E}$ .

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(III) (a) When a circular parallel-plate capacitor is being charged as in Example 31–1, show that the Poynting vector S→\(\overrightarrow{S}\) points radially inward toward the center of the capacitor, parallel to the plates.

Verified video answer for a similar problem:

Key Concepts

Poynting Vector

Electric Field in Capacitors

Magnetic Field and Charging

(III) (a) When a circular parallel-plate capacitor is being charged as in Example 31–1, show that the Poynting vector $\vec{S}$ points radially inward toward the center of the capacitor, parallel to the plates.