Why are E and B field in phase in far-field electromagnetic wave propagation?

I am very frustrated with the fact that the E and B field of an EM wave are in phase in the far-field case.

Before I get into this question I am letting you know that I already read this link related to my question, but it didn't really help me so much.

First, I know that an EM wave propagates as shown in the GIF below.

But, how are E and B field in phase? According to Maxwell's equation #3 and 4, we know that the change of E field induces B field, and vice versa. I also attached these equations below. I deleted the J term since we are talking about the far-filed analysis which lets us ignore the B field induced by the current flowing, J, in the wire.

Let's assume that the wave propagates along the x-axis. According to Maxwell, when there is a maximum E filed occurs (dE(t)/dt = 0) at a given point, x0, a minimum B field (B=0) should occur at the point. This should result in 90° phase difference between E and B field.

I know that we can calculate and prove that the EM wave propagates at a speed of light from Maxwell's equation #3 and 4 as shown below (source).

But the problem is that I don't know how to solve partial derivative equations (PDEs), and also I want some visualizations to understand this phenomenon better.

Is there any easy way to visualize why it is like this without solving PDEs?

Thank you.

• "Let's assume that the wave propagates along the x-axis. According to Maxwell, when there is a maximum E filed occurs (dE(t)/dt = 0) at a given point, x0, a minimum B field (B=0) should occur at the point. This should result in 90° phase difference between E and B field." A moment when both waveforms have a slope of zero at the same time doesn't sound like 90 degrees out of phase to me. If anything that's either in phase or 180 degrees out of phase. – DKNguyen Mar 23 at 19:44
• physics.stackexchange.com/questions/181277/… – Sredni Vashtar Mar 24 at 1:00
• Hi, thank you for your answer to my question. In my example, when the slope of E is 0 and the magnitude of B is 0 (not the slope of it). The slope of B would be higher or lower than 0 which means that E and B are out of phase, but they have to be in phase according to Maxwell... – Young Soung Park Mar 25 at 5:25
• Sredni Vashtar, thank you for your link. I looked at it and it helped me understanding my issue a bit better. Though, I still don't understand fully. – Young Soung Park Mar 25 at 5:36

The impedance of free space is not a complex impedance quantity; it is resistive with a value of $$\120\pi\$$ or about 377 ohms. This means that the magnitude ratio of E to B is 377:1 and, each are linked in phase to one another due to that non-complex resistance. If you expect a full derivation of Maxwell's equations on this site, you are expecting too much - try stack exchange physics.