Show that the mean power radiated by an oscillating dipole of amplitude \(p_{6}\) is given by \((14.52)\).

An oscillating electric dipole is a fundamental concept in electromagnetism, representing a pair of equal and opposite charges vibrating back and forth. The electric field created by such a dipole plays a significant role in the radiation of energy. Specifically, at any given point in space, this field arises from the time-varying separation of the charges and can be described mathematically by the following equation:

\[\begin{equation}\vec{E}(\vec{r},t) = k \cdot \frac{|\vec{p}(t)|}{4 \pi r^3} \left [ \vec{r} - (\vec{r} \cdot \hat{p}) \hat{p}\right]\end{equation}\]
Here, \(k\) is the electrostatic constant, \(\vec{p}(t)\) is the time-varying dipole moment, and \(\vec{r}\) is the position vector from the dipole to the observation point. Essentially, the oscillation of the dipole moment generates an electromagnetic wave that can transport energy through space, an effect that is especially crucial in the fields of communications and physics research.

Accompanying the electric field, the oscillating electric dipole also induces a magnetic field. This arises due to the movement of charges and their associated current within the dipole. According to Ampere's Law, the magnetic field caused by an oscillating electric dipole can be calculated using the cross product of the position vector and the time-dependent dipole moment:

\[\begin{equation}\vec{B}(\vec{r},t) = \frac{\mu_0}{4 \pi} \frac{1}{r^3} \vec{p}(t) \times \vec{r}\end{equation}\]
The magnetic field \(\vec{B}\) is essential for the formation of electromagnetic waves, which consist of oscillating electric and magnetic fields. In the case of an oscillating dipole, the interplay between these fields is what allows it to emit radiation that propagates through space.

Understanding the Poynting vector is crucial for grasping how energy is transmitted by electromagnetic waves. The Poynting vector, denoted as \(\vec{S}\), represents the directional energy flux (the rate of energy transfer per unit area) carried by an electromagnetic wave. For an oscillating dipole, the Poynting vector is given by the cross-product of the electric and magnetic fields:

\[\begin{equation}\vec{S}(\vec{r},t) = \frac{1}{\mu_0} \vec{E}(\vec{r},t) \times \vec{B}(\vec{r},t)\end{equation}\]
This vector is crucial for understanding how energy moves through space and is a key concept in the study of electromagnetic radiation. Knowing the Poynting vector's magnitude and direction enables one to determine the flow of energy from the oscillating dipole into the surrounding environment.

The radiated power by an oscillating dipole refers to the total energy emitted as electromagnetic radiation per unit time. By integrating the Poynting vector over a closed surface enclosing the dipole, such as a sphere with radius \(r\), we can determine the instantaneous power emitted at any given moment. The equation for this is:

\[\begin{equation} P(t) = \oint \vec{S}(\vec{r},t) \cdot d\vec{A}\end{equation}\]
To find the mean power radiated over one cycle of oscillation, we take the time average of the instantaneous power, which yields:

\[\begin{equation}\bar{P} = \frac{1}{T} \int_{0}^{T} P(t) dt\end{equation}\]
This mean power is critical for various practical applications, such as antenna design in telecommunications, where engineers need to know the average energy output over time to ensure the efficient operation of the system.