Animated Solution for Physics - Electromagnetic Waves: In free space, the energy of electromagnetic wave in electric field is $U_E$ and in magnetic field is $U_B$. Then

Imagine a wave of pure energy traveling through the vast emptiness of space. This is an electromagnetic wave. It is not made of matter, but rather of two intertwined partners: an electric field $\vec{E}$ and a magnetic field $\vec{B}$.

These fields are locked in a perpetual dance, oscillating perpendicular to each other and to the direction they are moving. But a profound question arises: how is the energy of this wave distributed between these two fields? Does one carry more energy than the other?

To answer this, we must look at the energy stored per unit volume, known as the energy density.

For the electric field, the energy density $U_E$ is given by the formula: where ${\epsilon }_0$ is the permittivity of free space.

Similarly, the energy density $U_B$ for the magnetic field is: where ${\mu }_0$ is the magnetic permeability of free space.

To compare $U_E$ and $U_B$, we need a way to connect the electric field $E$ and the magnetic field $B$. This is where James Clerk Maxwell's brilliant equations come to our rescue.

Maxwell showed that the speed of light $c$ in a vacuum is intimately tied to the constants of free space:

Furthermore, he proved that the amplitudes of the electric and magnetic fields in an electromagnetic wave are strictly proportional:

Now, let's find the ratio of the electric energy density to the magnetic energy density. We simply divide $U_E$ by $U_B$:

The factor of $\frac{1}{2}$ cancels out beautifully. Rearranging the terms, we get:

Now, we substitute our Maxwell relations. We know that ${\mu }_0{\epsilon }_0=\frac{1}{c^2}$ and $\frac{E^2}{B^2}=c^2$. Let's plug these in:

Look at that! The $c^2$ terms cancel out perfectly, leaving us with exactly $1$.

This simple result carries a profound physical meaning. Because the ratio is exactly $1$, it means that:

In any electromagnetic wave traveling through free space, the energy is perfectly and equally shared between the electric and magnetic fields. It is a beautiful symmetry of nature.

Because of this equal sharing, the total energy density $U$ of the wave can be expressed simply as twice the electric energy density, or twice the magnetic energy density: