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Maxwell’s equations for electromagnetic waves

Physicists illustrate EM waves as perpendicular E and M waves. That seems consistent. What does not make sense is the illustrations showing the E wave phase equal to the M wave. This seems to violate Maxwell’s equations.
The largest E potential would seem to be when the M wave is at a maximum of di/dt when the M wave crosses the zero axis. When maximum changes to the E wave magnitude occur as the E wave magnitude crosses the 0 axis then would not this correspond to the maximum in the M wave?
The wave nature of electromagnetic waves, as a consequence of Maxwell’s equations, makes sense except it would appear that the E and M waves should have a 90 degree separation. What am I missing?

John Varga asked

The following links may help you

http://physics.bu.edu/~duffy/PY106/EMWaves.html

Electromagnetic Waves and their Properties

http://farside.ph.utexas.edu/teaching/302l/lectures/node119.html

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