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

Verified step by step guidance

1. Step 1: Understand the Poynting vector and its role in energy transfer. The Poynting vector \( \mathbf{S} \) is defined as \( \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} \), where \( \mathbf{E} \) is the electric field, \( \mathbf{B} \) is the magnetic field, and \( \mu_0 \) is the permeability of free space. It represents the energy flux (rate of energy transfer per unit area) in an electromagnetic field.
2. Step 2: Analyze the fields in the capacitor. When the capacitor is being charged, there is a time-varying electric field \( \mathbf{E} \) between the plates and a magnetic field \( \mathbf{B} \) in the region around the capacitor due to the displacement current. The magnetic field \( \mathbf{B} \) is circular around the axis of the capacitor, and the electric field \( \mathbf{E} \) is perpendicular to the plates.
3. Step 3: Determine the direction of \( \mathbf{S} \). Using the cross product \( \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} \), the Poynting vector points radially inward toward the center of the capacitor. This is because \( \mathbf{E} \) is directed perpendicular to the plates (along the axis), and \( \mathbf{B} \) is circular around the axis, resulting in \( \mathbf{S} \) being radial.
4. Step 4: Set up the integral of \( \mathbf{S} \) over the cylindrical boundary. To calculate the total energy flux into the capacitor, integrate \( \mathbf{S} \) over the cylindrical surface enclosing the capacitor gap. The surface integral is \( \int \mathbf{S} \cdot d\mathbf{A} \), where \( d\mathbf{A} \) is the outward normal area element of the cylindrical surface.
5. Step 5: Relate the energy flux to the rate of energy storage. The rate at which energy enters the capacitor through the Poynting vector is equal to the rate of change of the electrostatic energy stored in the capacitor. The stored energy in the capacitor is \( U = \frac{1}{2} C V^2 \), where \( C \) is the capacitance and \( V \) is the voltage. Differentiate \( U \) with respect to time to find the rate of energy storage, and show that it matches the integral of \( \mathbf{S} \).

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 into the capacitor.

Capacitor Charging Process

When a capacitor is charged, an electric field is established between its plates, leading to the storage of electrostatic energy. The charging process involves the movement of charge carriers, which creates a potential difference across the plates. This process can be analyzed using Maxwell's equations, which describe how electric and magnetic fields interact, and is essential for understanding the dynamics of energy flow in the capacitor.

Energy Storage in Electric Fields

The energy stored in the electric field of a capacitor is given by the formula U = (1/2)CV², where C is the capacitance and V is the voltage across the plates. This energy is a result of the work done to separate charges against the electric field. Understanding how this energy relates to the Poynting vector and the rate of energy transfer is key to demonstrating that the energy entering the capacitor matches the energy stored in its electric field.