Animated Solution for Physics - Electromagnetic Waves: For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric ($U_e$) and magnetic ($U_m$) fields is

Imagine an electromagnetic wave travelling through the vast emptiness of free space. As it propagates forward, it isn't just an abstract mathematical concept; it is actively carrying energy. This energy is not localized in a single particle but is continuously shared and distributed between two oscillating entities: the electric field ($\vec{E}$) and the magnetic field ($\vec{B}$).

To truly understand how this energy is distributed, we must quantify it. In physics, when dealing with fields spread over space, we use the concept of energy density, which is simply the amount of energy stored per unit volume.

The average energy density stored in the electric field, denoted as $U_e$, is given by the fundamental relation:

Here, ${\epsilon }_0$ is the electrical permittivity of free space, and $E$ is the amplitude of the electric field.

Similarly, the average energy density stored in the magnetic field, denoted as $U_m$, is expressed as:

In this equation, ${\mu }_0$ represents the magnetic permeability of free space, and $B$ is the amplitude of the magnetic field.

At first glance, $U_e$ and $U_m$ look like entirely different beasts. One depends on ${\epsilon }_0$ and $E$, while the other depends on ${\mu }_0$ and $B$. How can we possibly compare them?

The secret lies in the fundamental nature of electromagnetic waves. The electric and magnetic fields do not exist independently; they are intimately coupled. The magnitude of the electric field is directly proportional to the magnitude of the magnetic field, linked by the speed of light $c$:

Furthermore, the speed of light itself is not an arbitrary number. It is deeply woven into the fabric of space, defined by the permittivity and permeability:

Now, let's perform a mathematical substitution to reveal the hidden symmetry. We can express the electric field $E$ entirely in terms of the magnetic field $B$ and the constants of free space:

Let's carefully substitute this expression back into our equation for the electric energy density $U_e$:

When we square the term inside the parentheses, the square root in the denominator vanishes:

Look closely at what happens next. The ${\epsilon }_0$ in the numerator perfectly cancels out the ${\epsilon }_0$ in the denominator. This is not a coincidence; it is the mathematical manifestation of the wave's symmetry. We are left with:

Take a moment to look at our final expression for $U_e$. It is exactly identical to the formula for the magnetic energy density $U_m$!

This is a profound and beautiful result. It tells us that in a propagating electromagnetic wave, the energy is perfectly equipartitioned. The electric field and the magnetic field carry exactly the same amount of energy. Consequently, the total energy density of the wave is simply the sum of both, which is twice the electric energy density or twice the magnetic energy density.