Electromagnetism – Are E and B Fields in Phase in Electromagnetic Radiation?


I recently wrote this answer, in which I said:

Radio waves are electromagnetic radiation. Electromagnetic radiation contains of two components, one electrical and one magnetic. These components create each other, as said above. The red magnetic field creates a blue electric field, which creates the next magnetic field, and so on.

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I got this diagram from wikipedia, but my physics book and Jim Hawkins WA2WHV give the same diagram.

In the comments, a discussion followed:

Olin Lathrop: Your first diagram is wrong. The B and E fields are actually 90 degrees out of phase with each other, not in phase as the diagram shows. The energy is constantly sloshing back and forth between the E and B fields.

Keelan: Are you sure? Wikipedia and my physics book show different. The two fields should have a fixed ratio, I believe, which cannot happen when out of phase. One field is horizontal and the other vertical, there's a 90 degrees angle – the diagram is an attempt to show three dimensions.

Olin Lathrop: Hmm. I always understood them to be in quadrature, but I don't have time to look that up right now. This could be a case of one bad diagram blindly copied by lots of others. Where is the energy when both fields hit 0 in your diagram? In quadrature, the sum of the squares of each field's amplitude is a constant, which gives a good explanation of how the energy can persist. It sloshes back and forth between the two fields, but its total is always the same.

I follow Olin's logic and can't say myself why the fields would be in phase. So my question is: are the E and B fields of electromagnetic radiation in phase or not? How can one understand this?

Best Answer

The complete derivation from Maxwell's equations fills entire college-level textbooks, and is too involved to get into here.

But when considering radiation from an antenna (a current flowing in a linear conductor), it boils down to the fact that there are several distinct components to both the E (electric) and H (magnetic) fields around the antenna. For the H field, there is one component that is proportional to 1/r2 and another that is proportional to 1/r. For the E field, there are three: a 1/r3 component, a 1/r2 component and a 1/r component.

The 1/r3 term is the dipole electrostatic field, which represents the energy stored in a capacitive field. Similarly, the 1/r2 term represents the energy stored in an inductive field. This represents the "self inductance" of the antenna conductor, in which the magnetic field produced by the current induces a "back EMF" on the conductor itself. Only the 1/r term represents energy that is actually carried away from the antenna.

Near the antenna, where the 1/r3 and 1/r2 components dominate, the phase relationship between E and H is complicated, and these fields do indeed store energy in the manner that Olin describes, and return energy back to the antenna itself.

However, in the "far field" (e.g., more than 10 wavelengths away from the antenna), the 1/r components of the fields dominate, creating the propogating electromagnetic plane wave, and these components are indeed in phase with each other.

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