confused by explanation of electromagnetic wave in physics textbook
I'm kind of confused by the proof used in my physics textbook to explain an electromagnetic field using Faraday's law.
The text book uses Faraday's law to try to explain why a uniform electric field pointing in the y direction while moving in the x direction must have a uniform magnetic field perpendicular to it pointing in the z direction. The images provided are:
The explanation goes on to say that since the application of Faraday's law to the change in size by dA of the imaginary boundary "efgh" as the uniform electric field moves in the x direction at the speed of light c for time dt provides a value of $$\oint {efgh}\overline{E}\cdot d\overline{l}=-Ea$$ for the lefthand side of <PERSON>'s law (since a is the distance from g to h in the image provided and E is parallel to and exists within only that side of the imaginary boundary and in the opposite direction) which is a nonzero value, it must mean that the righthand side of <PERSON>'s law $$-\frac{d\phi {B}}{dt}$$ must be nonzero as well. 903
This basically tells us that there must be a uniform magnetic field pointing in the z direction perpendicular to the uniform electric field pointing in the y direction.
While I do understand this explanation, I'm kind of confused because what if the boundary "efgh" was drawn like this instead?
If this was the case, from my understanding the lefthand side of <PERSON>'s law $$\oint {efgh}\overline{E}\cdot d\overline{l}=0$$ because the electric field is parallel and exists on (g to h) and (e to f) of the imaginary boundary "efgh," but since it is opposite in direction to the (g to h) side and in the same direction as the (e to f) side, they should cancel out one another. But wouldn't there still be a change in the magnetic flux through the area dA? So it should mean that the righthand side of <PERSON>'s law $$-\frac{d\phi {B}}{dt} \neq 0$$ But this seems to contradict <PERSON>'s law since $$\oint {efgh}\overline{E}\cdot d\overline{l}=0=-\frac{d\phi {B}}{dt} \neq 0$$ I'm sure there must be something wrong with this approach. What is it that I'm missing? Any ideas? 903