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

Verified step by step guidance

1. Identify the electric field ($\overrightarrow{E}$) direction: In a charging capacitor, the electric field between the plates is directed from the positive plate to the negative plate, perpendicular to the plates.
2. Identify the magnetic field ($\overrightarrow{B}$) direction: According to the right-hand rule, the magnetic field lines are circular and concentric around the wire carrying the current to the plates, with the direction depending on the direction of the current.
3. Understand the Poynting vector ($\overrightarrow{S}$): The Poynting vector is given by $\overrightarrow{S}=\frac{1}{{\mu }_0}(\overrightarrow{E}\times \overrightarrow{B})$, where ${\mu }_0$ is the permeability of free space. This vector represents the direction of energy flow.
4. Calculate the cross product ($\overrightarrow{E}\times \overrightarrow{B}$): Since $\overrightarrow{E}$ is perpendicular to the plates and $\overrightarrow{B}$ is tangential (circular around the wire), the cross product $\overrightarrow{E}\times \overrightarrow{B}$ will point radially, either inward or outward depending on the direction of $\overrightarrow{B}$ which is determined by the current direction.
5. Conclude the direction of $\overrightarrow{S}$: For a typical setup where the current flows into the capacitor through the central wire, the right-hand rule for the cross product will show that $\overrightarrow{S}$ points radially inward, indicating that energy is flowing into the capacitor towards the center.

Key Concepts

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, the electric field (→E) is uniform and directed from the positive plate to the negative plate. The strength of the electric field is given by E = V/d, where V is the voltage across the plates and d is the separation between them. This electric field is essential for understanding how charges accumulate on the plates and how energy is stored in the capacitor.

Magnetic Field and Charging

When a capacitor is charged, a changing electric field generates a magnetic field (→B) around it, according to Maxwell's equations. The magnetic field is circular and perpendicular to the electric field lines, and its direction can be determined using the right-hand rule. This interplay between electric and magnetic fields is fundamental to the behavior of electromagnetic waves and is key to analyzing the Poynting vector in the context of a charging capacitor.