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:

In an electromagnetic wave, the energy is shared equally between the electric and magnetic fields. The average energy density due to the electric field is $$U_e = \frac{1}{2} \varepsilon_0 E_{\text{rms}}^2 = \frac{1}{4} \varepsilon_0 E_0^2$$, and the average energy density due to the magnetic field is $$U_m = \frac{B_{\text{rms}}^2}{2\mu_0} = \frac{B_0^2}{4\mu_0}$$.

Since $$E_0 = cB_0$$ and $$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$$, we have $$U_e = \frac{1}{4} \varepsilon_0 c^2 B_0^2 = \frac{1}{4} \varepsilon_0 \cdot \frac{1}{\mu_0 \varepsilon_0} \cdot B_0^2 = \frac{B_0^2}{4\mu_0} = U_m$$.

Therefore, for an electromagnetic wave travelling in free space, the average energy densities due to the electric and magnetic fields are equal: $$U_e = U_m$$.