Total Energy of an electromagnetic wave

Question

This question may sound stupid. But why when we calculate the total energy density (energy/volume), carried by an electromagnetic wave $u_T$, we add both $u_B$ + $u_E$.
Where $u_B = \frac{B^2}{2\mu_0}$ and $u_B = \frac{\epsilon_0 E^2}{2}$
From what I understand (Which I feel is wrong), the magnetic field cannot do work, and thus it transfers its energy to another field, induced Electric Field, in order to do work, this question can further explain my point here.
Then my question is if in electromagnetic waves the E field is induced by the changing B field and vice versa, wouldn't that mean that the energy we refer to as $u_B$ the same as $u_E$ and thus making the sum of them as summing the same thing twice?
Note: I'm not yet into quantum mechanics, I know everything changed there, but I have not yet studied it.

Answer 1

This comes right from Maxwell's equations as follows:

There is a vector identity that $$(\nabla \times \vec E) \cdot \vec B -(\nabla \times \vec B) \cdot \vec E = \nabla \cdot (\vec E \times \vec B)$$ This is a vector identity so it holds for all vector fields. Now, if we substitute in the microscopic forms of Faraday's law and Ampere's law (in natural units to avoid all of the constants) then we get $$-\vec B \cdot \frac{\partial}{\partial t} \vec B - \vec E \cdot \vec J - \vec E \cdot \frac{\partial}{\partial t}\vec E= \nabla \cdot (\vec E \times \vec B)$$ $$ 0 = \frac{1}{2} \frac{\partial}{\partial t} E^2 + \frac{1}{2} \frac{\partial}{\partial t} B^2+ \nabla \cdot (\vec E \times \vec B) + \vec E \cdot \vec J$$ $$0=\frac{\partial}{\partial t} u + \nabla \cdot \vec S + \vec E \cdot \vec J $$ where $u=\frac{1}{2}(E^2+B^2)$ is interpreted as the microscopic electromagnetic field energy density, and $\vec S = \vec E \times \vec B$ is interpreted as the microscopic electromagnetic field energy flux density. You cannot get rid of the $B^2$ term since it falls right out of Maxwell's equations.

It may help to think of a loop of superconducting wire, like a MRI magnet. There is a very large current with a very large magnetic field which has a very large energy density as described above. Because it is superconducting there is no voltage when the current is steady. However, in an emergency the field can be quenched. When this happens the large amount of energy in the field gets dissipated into the wire and then into the liquid helium and liquid nitrogen. This causes a rapid boil-off of the cryogens and is very loud and dramatic when it happens. Since boiling the cryogens that quickly requires a substantial amount of energy and since the only source of that energy was the magnetic field, then it is clear that the magnetic field energy is a real thing, as suggested by the theory, and cannot be neglected.