Prove that, in an EM wave traveling in vacuum, the electric...

Prove that, in an EM wave traveling in vacuum, the electric and magnetic energy densities are equal; that is, prove that
$$
\frac{1}{2} \epsilon_{0} E^{2}=\frac{1}{2 \mu_{0}} B^{2}
$$
at any point and at any instant of time.

Key Concepts

Relationship Between Electric and Magnetic Fields in Vacuum

In a vacuum, a key result of Maxwell’s equations is that the magnitudes of the electric and magnetic fields in an electromagnetic wave are directly related by the speed of light, given by E = cB, where c = 1/?(????). This proportionality not only ensures that the wave propagates at a constant speed but also equally distributes energy between the electric and magnetic components. Proving that the energy densities are equal relies heavily on this intrinsic relationship.

Electromagnetic Wave Propagation in Vacuum

This concept refers to the manner in which electromagnetic waves travel through empty space (vacuum). In a vacuum, there is no medium to interact with the wave, and Maxwell’s equations predict that electric and magnetic fields oscillate perpendicularly to each other and to the direction of propagation at the speed of light, c. This propagation underpins the fundamental understanding of light and other forms of electromagnetic radiation.

Energy Density of Electromagnetic Fields

The energy density in an electromagnetic field quantifies the amount of energy stored per unit volume. It is divided into two parts: one related to the electric field (½??E²) and one related to the magnetic field (½(1/??)B²). This concept is critical because it shows how energy is partitioned within an electromagnetic wave, and it lays the foundation for analyzing energy transport and interactions in various physical contexts.

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