Apparent Violation of Energy Conservation in an Electromagnetic system

As shown in the figure a beam of EMW pulse is propagating towards a nearly point charge.Here if we calculate the total energy of the system by the equation mentioned in photo - we get that the total energy of the system increases as the beam moves towards the charge.But simply we are not supplying any energy to the system.Why apparently some extra energy is coming into the picture?

• You will have the same problem with two point charges approaching each other. That is why the concept of potential energy was invented. – anna v Jan 29 '15 at 15:00
• But we can explain the potential energy involved in that case in the terms of work done required to bring the charges together because Coulombic forces act on charges...In this case no force act on either of the charge or EMW – Dvij Mankad Jan 29 '15 at 15:09
• Your pulse was generated at some distance R. In the generation potentials were involved, though more complicated mathematically than two simple charges. The pulse will scatter off the charge and go in its deflected way. Energy will be conserved. – anna v Jan 29 '15 at 15:16
• Forget the generation of pulse.Theoretically we can say that pulse was there.No need to get it generated.Now the pulse will get scattered off the charge when it passes through the charge but i am getting problem even before the pulse reaches the charge. – Dvij Mankad Jan 29 '15 at 15:23
• You cannot forget the generation, you put it there, at the R point (vector). A ball falling increases its kinetic energy. You cannot say it existed at R over the ground. See farside.ph.utexas.edu/teaching/em/lectures/node89.html . Search for "gain or lose" – anna v Jan 29 '15 at 15:28

No, it is not. As we know: $$\iint_{S} (\mathbf{E} \cdot d\mathbf{S})=4\pi Q$$ If we take a small box located at the edge of a beam as a surface of integration, we get to the conclusion that some charge must be placed there, because there is electric field inside the beam and no field outside. So, electric field can't be homogenous inside of it.