I've learnt quantum field theory for a semester but I still can't understand the role of classical equation of motion in QFT. I have looked up for several books. They all discuss classical field theory. And they turn to the quantum part without leaving comments. It seems the following things have...

Consider two parallel plate capacitors, one with a separation between plates of $d_1$ and other with $d_2$, where $d_1,d_2<<\sqrt A$ and where $A$ is the area between the plates and both in the same medium. Now let dielectric strength of material be $E$, therefore the maximum possible electric fi...