On the topic of plane EM waves, can someone please explain the whole deal why th

Question

On the topic of plane EM waves, can someone please explain the whole deal why there is flux on the left hand side but not on the right if there is a component of E or B parallel to the direction of propagation?

Let us first verify that our wave satisfies Maxwel's first and second equations_that is, Gauss's laws for electric and magnetic fields. To do this, we take as our Gaussian surface a rectangular box with sides parallel to the xy, xz and yz coordinate planes (Fig. 32.6). The box encloses no electric charge. The total electric flux and magnetic flux through the box are both zero, even if part of the box is in the region where EB0. This would not be the case if E or B had an x-component, parallel to the direction of propagation; if the wave front were inside the box, there would be flux through the left-hand side of the box (at x = 0) but not the right-hand side (at x 0). Thus to satisfy Maxwell's first and second equations, the electric and magnetic fields must be perpendicu- lar to the direction of propagation; that is, the wave must be transverse.

Explanation / Answer

Let's say wave is propagating in x-direction and it is continously propagating, but at each time whenever you are observing wavefront would be inside the box. Let's say electric field has a component in x direction. Then this wavefront will have electric field component which is parallel to x axis, this has entered inside the box from left hand side and is still inside the box, hence flux is there inside the box. Now as the EM wave is continously propagating, now this wavefront will leave the system and other will enter, again at every time you observe there exists some flux inside the box, which has entered from x<0 but hasn't existed from x>L.

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