Energy Conservation in electromagnetism

Question

I have some trouble accepting the Conservation of Energy in electromagnetism. Assume two EM pulses in opposite directions move toward each other and are entirely in reflection symmetry (their field vectors are like $\vec{E}$ and $-\vec{E}$ in all time). when they interfere, there is a moment they should completely cancel each other and there should be $\vec{E}=\vec{0}$ in all space, and according to $$\text{Energy}=\int{\epsilon _0 E^2}\mathrm{d}V$$ the energy in the entire space is zero but before interference, we had energy (sum of each pulse energy). so how conservation of energy is valid here? (we can also express everything I've said about magnetic field too)

Answer 1

we can also express everything I've said about magnetic field too

Here is your mistake. In fact, in the situation you have described the magnetic fields must add constructively.

Recall that for an EM wave we have $\vec S = \vec E \times \vec H$ where $\vec S$ defines the direction of propagation. Thus, in your case, since the pulses are moving in opposite directions, we must have $\vec S_1 = -\vec S_2$. So $$\vec E_1 \times \vec H_1 = - \vec E_2 \times \vec H_2$$ And since you have further specified that $\vec E_1=-\vec E_2 = \vec E$ we have $$\vec E \times \vec H_1 = \vec E \times \vec H_2$$$$\vec H_1=\vec H_2$$

So in the very instant when the E fields cancel out the H fields align. The complete destructive interference of the E fields is balanced by the complete constructive interference of the H fields.

Furthermore, in an EM wave half of the energy is in the E field and half of the energy is in the H field, so the total energy of the E and H fields of both waves is 4 times the energy in the E field of one wave. Also, the energy is proportional to the square of the field, so when the E fields cancel out, the H field is doubled, so the energy in the combined H field is 4 times the energy in the E field of one wave, thus conserving energy.

EDIT: in response to @hyportnex comments below I work this out for a specific example using units where $\mu_0=\epsilon_0=c=1$ for simplicity.

Let $$u(x)=\begin{cases} 1 & -\frac{1}{2}<x<\frac{1}{2} \\ 0 & \text{otherwise} \end{cases}$$

Now, suppose $$\vec E_1 = u(x-t) \hat y$$$$\vec H_1 = u(x-t) \hat z$$and$$\vec E_2 = - u(x+t) \hat y$$$$\vec H_2 = u(x+t) \hat z$$You can confirm that these fields satisfy the OP's requirements for two identical waves propagating in opposite directions. You can also confirm that they satisfy Maxwell's equations.

Now, let$$\vec E=\vec E_1+\vec E_2$$$$\vec H = \vec H_1 + \vec H_2$$You can also confirm that this satisfies Maxwell's equations, although it should be obvious by the linearity of Maxwell's equations without doing a lot of math. Nevertheless, you can also confirm that $\left. \vec E\right|_{t=0}=0$ everywhere, which satisfies the OP's requirements that the E field cancel out completely. You can further confirm that $\left. \vec H \right|_{t=0}=2 \left. \vec H_1\right|_{t=0} = 2\left. \vec H_2\right|_{t=0}$ as I stated above.

So in this example, as described in general above, as claimed when the E fields add destructively the H fields add constructively and the energy is conserved.