Show that the mean electric and magnetic energies in a plane electromagnetic wave in vacuo are equal.

Electromagnetic waves are ripples of energy that travel through the vacuum of space at the speed of light, characterized by oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. These fields are created by accelerating charges and do not require a medium to travel through, differentiating them from mechanical waves.

The energy carried by an electromagnetic wave is distributed between its electric and magnetic fields. This energy distribution is crucial to applications ranging from radio broadcasting to microwave heating. Understanding how energy is stored in these fields opens the door to innovations in communication, medical technology, and many other fields.

The electric field amplitude, denoted as E or E₀ for its maximum value, represents the highest strength of the electric field in an electromagnetic wave. It dictates the amount of force that would be exerted on a charge placed within the field. The energy density of the electric field is proportional to the square of this amplitude, signifying that even small increases in electric field amplitude can lead to significant amounts of energy stored within the field.

This concept helps explain phenomena such as the intensity of light: higher electric field amplitude corresponds to brighter light, and in the context of antennas, it means more powerful transmission of signals. For students, grasping the implication of E's amplitude on the energy density is key to understanding wave behaviors.

In tandem with its electric counterpart, the magnetic field amplitude, symbolized by B or B₀ at its peak, indicates the maximum magnitude of the magnetic field in the wave. This value, while often less intuitive than its electric sibling, is integral to the magnetic aspect of energy storage in an electromagnetic wave. In a vacuum, the energy density of the magnetic field, just like the electric component, is proportional to the square of the magnetic field amplitude.

The concept that a magnetic field stores energy and does work is a leap from the classic iron-filings visual of magnetic lines of force. The realization that a changing magnetic field is fundamental to the operation of generators and motors underlies much of modern technology. When students understand the magnetic field's role alongside the electric field, they unlock a complete picture of electromagnetic wave dynamics.