Show that the time-average Poynting vector for the far fields (8.9.13) of an oscillating electric dipole \((8.9 .12)\) is $$ \underline{E}=\hat{\mathrm{f}} \frac{c k^{4} p_{0}^{2}}{32 \pi^{2} \epsilon_{0}} \frac{\sin ^{2} \theta}{r^{2}} $$ and that the average total power radiated by the oscillating dipole is $$ \left(\frac{d W}{d l}\right)=\frac{c \kappa^{4} p_{0}^{2}}{12 \pi \epsilon_{0}} $$ Why is the sky blue and the sunset red?

An oscillating electric dipole consists of two charges of equal magnitude but opposite signs, separated by a small distance, oscillating in opposite directions. As this occurs, the dipole moment, which is a vector pointing from the negative to the positive charge, changes in magnitude and direction over time. This dynamic variation of the dipole moment creates an oscillating electric field and an associated magnetic field, allowing the dipole to radiate electromagnetic waves.

In technical terms, an oscillating electric dipole can be represented as a time-varying function, where the dipole moment \(p_0\) oscillates with a dependence on the angular frequency \(\omega\), which is related to the frequency of oscillation. The radiation pattern of an oscillating electric dipole is defined by its far fields, which diminish with distance but can extend to large scales.

The characteristics of the fields produced by such a dipole are essential in understanding how antennas radiate and receive signals and in explaining various physical phenomena, such as Rayleigh scattering, which influences the color we perceive in the sky.

The time-average power radiated by an oscillating electric dipole refers to the mean power output over one complete cycle of oscillation. It is an essential quantity as it describes the energy emitted into the surrounding space in the form of electromagnetic radiation.

Mathematically, this power is related to the Poynting vector \(\underline{S}\), which represents the flow of electromagnetic power per unit area. By calculating the time average of the Poynting vector for the fields far from the dipole and then integrating over all directions, we obtain the total power radiated by the dipole.

This concept helps us understand how much energy is being lost from the system to the environment and serves as a fundamental property in designing and understanding antennas, which must often maximize the amount of power radiated into specific directions for efficient communication.

Rayleigh scattering is an optical phenomenon that occurs when light or other electromagnetic radiation interacts with particles that are much smaller than the wavelength of the radiation. The shorter wavelengths of light (bluer) are scattered more effectively compared to the longer wavelengths (redder) due to their greater interaction with small particles and molecules in the atmosphere.

This scattering can explain various natural occurrences like the color of the sky. As sunlight enters Earth's atmosphere, it encounters gases and tiny particles that scatter the shorter blue and violet wavelengths more than the longer red ones, which is why the sky appears blue during daylight hours.

At sunrise and sunset, the sun is at a lower angle in the sky, and the sunlight has to pass through more of the Earth's atmosphere to reach an observer. During this time, much of the blue light is scattered out of the direct path between the sun and the observer, allowing the longer wavelengths, which are scattered less, to become more pronounced and give the sky its beautiful red and orange hues during these periods. Understanding Rayleigh scattering is also crucial for various applications, including remote sensing, astrophysics, and telecommunications.