In a vacuum, the plane wave solutions to Maxwell's Equations are... E=E0*cos(wt-kr) B=B0*cos(wt-kr) ie they are in phase. (See for example https://www.physics.wisc.edu/undergrads/courses/spring08/208/Lectures/lect20.pdf http://hyperphysics.phy-astr.gsu.edu/hbase/waves/emwv.html ) I don't understand how they can be in phase. I expected them to be 90 degrees out of phase because a changing electric field causes a magnetic field. So when the electric field is changing the most the magnetic field should be greatest. Since the differential of cos is sin, I'd expected the magnetic field to be 90 degrees out of phase. Where am I going wrong?