In an electromagnetic wave, why are the electric and magnetic fields both at their maximum at the same time and at the same place?
The fact that the electric and magnetic fields are "in phase" with one another is a counterintuitive result.
After all, Maxwell’s equations state that it is the "rate of change" of the electric field that induces the magnetic field.
And the "rate of change" of the electric field is at a maximum when the electric field is zero.
Similarly, Maxwell’s equations state that it is the "rate of change" of the magnetic field that induces the electric field.
And the "rate of change" of the magnetic field is at a maximum when the magnetic field is zero.
So, why are the electric and magnetic fields not out of phase?
The key to the answer is that Maxwell’s equations relate the rate of change of each field to the curl of the other field.
The blue arrows describe a vector field, which we will call "F."
The orange vector is the curl of "F."
As the vector field changes, the curl changes as shown.
Let’s set two of the four field vectors to zero.
As the two remaining field vectors change, the curl changes as shown.
The curl is zero when the two neighboring field vectors are of the same length and are pointed in the same direction.
The curl is at a maximum when the two neighboring field vectors are pointed in opposite directions.
Let’s keep this in mind when analyzing the curl of the electric and magnetic fields of an electromagnetic wave.
Let’s take our electromagnetic wave, and consider just one moment in time.
The curl is at a maximum where the field is zero, which is where the neighboring fields to the left and to the right are in opposite directions.
The curl of the magnetic field is directly proportional to the rate at which the electric field is changing.
Let’s now consider the curl of the electric field.
The curl of the electric field is equal to the negative of the rate at which the magnetic field is changing.
Hence, the electric and magnetic fields are "in phase" with one another.
When people talk about fields being out of phase, they are typically talking about the addition of two different electromagnetic waves, out of phase with each other.
The fields from these two electromagnetic waves add together to produce a circularly polarized wave, which is another way that Maxwell’s Equations can be satisfied.
Much more information is available in the video "Polarization of Light", and in the other videos on this channel.
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