Show that $$ \begin{aligned} &\mathbf{E}=(1+j j) E_{1} e^{j(\omega t-\alpha o)} \\ &\mathbf{B}=(-1 j+j) \frac{E_{1}}{c} e^{j(\omega t-\alpha t)} \end{aligned} $$ represent a circularly polarised plane wave. (Note that \(j=\sqrt{-1}\), while \(\mathbf{1}, \mathbf{j}\) are the cartesian unit vectors in the \(x\) and \(y\) directions!) If you watch the time variation of the electric field at a fixed position, will the direction of the field rotate in the right-or left-handed sense with respect to the direction of travel \((+z) ?\) If you could take a snapshot of the electric field over space, in which sense would the direction rotate? Repeat these questions for the magnetic field. How would you represent a circularly polarized wave of the opposite handedness? Answer: Left-handed; right-handed; magnetic same as electric; reverse sign of \(j\) in coefficients.

Polarization is a fundamental characteristic of waves that describes the orientation of their oscillations. For electromagnetic waves, including light, this refers to the direction of the electric field's oscillation. There are several types of polarization, such as linear, circular, and elliptical. In linear polarization, the electric field oscillates in a single plane. However, for circular polarization, the electric field rotates in a circular manner as the wave propagates. This creates a helical pattern in the space through which the wave travels.

The key indicator of circular polarization is a constant phase difference between the two perpendicular components of the electric (or magnetic) field. For instance, in a right-handed circularly polarized wave, if you were to look along the direction of wave travel, the electric field vector would rotate clockwise. Conversely, it would rotate counterclockwise for a left-handed wave. Understanding these rotations and the polarization concept are crucial, not just for comprehending the fundamental properties of light, but also for applications such as optical filters and communication systems.

Electromagnetic waves are fascinating phenomena that encompass a wide range of frequencies, from radio waves to gamma radiation. These waves are characterized by their wavelength, frequency, speed, amplitude, and polarization. They travel at the speed of light in a vacuum (\( c \text{, which is approximately } 3 \times 10^8 \text{ meters per second}\)).

Their transverse nature means that the oscillations of the electric and magnetic fields are perpendicular to the direction of wave travel, as well as to each other. For a wave to be considered circularly polarized, its electric and magnetic field components should rotate with constant amplitude and maintain a consistent phase relationship. This is demonstrated in the textbook exercise, where the electric and magnetic fields of a circularly polarized wave are modeled with their corresponding mathematical representations.

The concept of phase difference plays a crucial role in the characterization of waves. Phase difference is the offset in the oscillation cycles of two waves or two components of a wave. It is typically measured in radians or degrees. When two components of a wave, such as the x and y components of an electromagnetic wave's electric field, have a phase difference of \( \pm \frac{\pi}{2} \) (or \( \pm 90^\circ \) for degrees), the wave is said to be circularly polarized.

In the provided exercise, the phase difference between the electric and magnetic fields is zero, implying a special condition where the fields could either be linearly polarized or part of a circularly polarized wave depending on their individual orientations and rotation. The significance of phase difference in engineering and physics cannot be understated, as it's critical in the synthesis and analysis of wave behavior in a multitude of applications.

Handedness, or chirality, in polarization refers to the direction in which the electric (or magnetic) field rotates as the wave propagates. The two types of handedness are right-handed and left-handed circular polarization. To determine the handedness of a wave, one can envision looking down along the wave's direction of travel. If the electric field rotates clockwise, the polarization is right-handed; if it rotates counterclockwise, the wave is left-handed.

This concept is particularly important when discussing the interaction of polarized light with materials, as certain substances can distinguish between left and right-handed polarization, a property employed in the fields of chemistry and biology to investigate molecular structures. As demonstrated in the exercise solution, the handedness can be manipulated by adjusting the phase and amplitude of the wave components, allowing for the creation of waves with the desired characteristics.