# 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. 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?

• "Where is the energy when both fields hit 0 in your diagram? ". Elsewhere. It's not like the E and B fields are momentarily 0 everywhere. – MSalters Apr 5 '13 at 19:01
• In the figure, Magnetic Field is in XY plane whereas the electric field is in YZ plane. (Assuming Z pointing upwards) So, isn't there a phase difference of 90 degree shown? Please correct if I'm wrong. – Enthusiast Jul 20 '13 at 9:56

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.

• I'd have liked to have seen a bit more of an explanation about 1/r^2 for both E and H fields. – Andy aka Apr 5 '13 at 16:38

The impedance of free space is constant. Its value is proportional to the ratio of E and H.

It is a resistive quantity which means E and H must rise and fall in magnitude together.

Wikipedia: - • This is the key ... I'll just add a little detail. E X B will still give a direction in the case of a time phase difference, the key point is that value will be complex (real and imaginary components) - i.e. have "storage". A purely real quantity will be resistive. – placeholder Apr 5 '13 at 14:17

The confusion stems from the fact that they (the instantaneous electric and magnetic vector fields) are 90 degrees apart in space, not in time. That is to say:

$E \cdot B = 0$ , and $E \times B$ is the direction of propagation (the poynting vector).

To quote the Wikipedia:

The electric and magnetic parts of the field stand in a fixed ratio of strengths in order to satisfy the two Maxwell equations that specify how one is produced from the other. These E and B fields are also in phase, with both reaching maxima and minima at the same points in space (see illustrations). A common misconception is that the E and B fields in electromagnetic radiation are out of phase because a change in one produces the other, and this would produce a phase difference between them as sinusoidal functions (as indeed happens in electromagnetic induction, and in the near-field close to antennas). • As a side note, the direction of these fields determines the polarisation of the signal. If you interchanged the axes of the E and B fields, some types of antenna would fail to pick up the signal until you rotated the antenna by 90 degrees. (Or some types of sunglasses would fail to transmit the signal) – Brian Drummond Apr 5 '13 at 14:32

Suppose we're given an electric field propagating in the $\hat{z}$ direction, $\vec{E} = \hat{x}E_0 \cos\left(\omega t - kz\right)$. Maxwell's curl equation relating the electric and magnetic fields is given by, $$\nabla \times \vec{E} = -\frac{\partial}{\partial t}\mu\vec{H}$$ This essentially relates the spatial derivatives of the electric field to the time derivative of the magnetic field. If we look at this equation, in order to find $\vec{H}$ we'll have to take the space derivative of $\vec{E}$ and then to find $\vec{H}$ we will have to integrate that same function over time, so essentially we end up with the same time-harmonic function we began with. Due to the negative sign on the time derivative of $\vec{H}$ we end up with the electric and magnetic fields being technically 180$^\circ$ out of phase, but they aren't in the same plane in space either.

Basically, diagrams like the one linked in the question can be nice for visualizing the fields in space, and if you look carefully you can see the field phasing. Looking at the equations can be just as revealing though, and if you go through the math Maxwell will give you the answer.

Voltage is not dependent on magnetic field but on its rate of change. Therefore induced voltage is highest when magnetic field is zero, when its derivative is highest.

For constant energy in an EM wave, we need the magnetic component and the electric component of voltage to be 90 degrees out of phase : thus we need the effect of the magnetic field to be greatest when the electric field is 0; this happens when the fields themselves are in phase.

• Aren't you confusing the induced voltage in an antenna with the E field of the electromagnetic wave? In vacuum, B=k̄/c☓E (with k̄ the wave direction) – MSalters Apr 5 '13 at 18:59