# Electromagnetic wave and the phase between the E and B fields

1. ### ANvH

54

According to the quoted thread above and according to textbooks and Wikipedia the phase between the E and B fields of an electromagnetic wave propagating in free space is zero. This assertion is based on the Maxwell equations using a planar wave.

DaleSpam rephrases Maxwell's laws in the quoted thread:

However, if the E-field is subject to an alternating displacement, the maximum and minimum field amplitudes occur when the rate of displacement is ##\frac{∂E}{∂t}=0##. According to the Maxwell equation
##∇ \times B=\frac{∂E}{∂t}##​
the curl of the magnetic field is then zero when the electric field is maximal or minimal. If the curl of the magnetic field is zero, then my interpretation of what the curl means, leads to the conclusion that the magnetic field itself has a zero value.
To continue with this, when the electric field is zero, the rate of change of the electric field is maximal and the curl of the magnetic field is maximal too, which leads to the conclusion that when the electric field does not exist (its rate of change is maxed), the magnetic field magnitude is maximal and rotational.

This goes against the interpretation of DaleSpam, which to me is solely based on a math interpretation, not a physical interpretation. I would conclude that there is a 90° phase difference and that therefore
##E=E_{0}sin(\omega t - kx)##​
is incompatible with the Maxwell equations. Of course I might be completely wrong, yet DaleSpam's explanation does not cut it for me, unless my interpretation of the curl of a vector field is out of touch.

2. ### UltrafastPED

1,912
For a correct analysis, please review http://www.phy.iitb.ac.in/~dkg/PH-102/emw.pdf

The E and B fields are in phase.

This is often used as a homework problem in your basic electromagnetic field theory course.

3. ### ANvH

54
Sorry, but I contest the planar wave as a solution. When using the planar wave then mathematically you should conclude that the E and B fields are in phase. When my homework is correct with respect to what the curl of a vector field means then I am at odds with the planar wave. Reviewing the pdf file you quote, it again uses the planar wave. It does not answer my question.

### Staff: Mentor

Does some incompatibility/contradiction appear if you substitute that expression for ##E## back into Maxwell's equations? And what expression do you get for the corresponding ##B## field?

That's the acid test for whether something is a valid solution for the equations. If you do that and you still find an incompatibility, show your work and one of us will be able to help you find the place where your calculation went astray.

5. ### UltrafastPED

1,912
So you don't have any issues with the mathematics, but perhaps with the existence of plane waves?

You could try solving for spherical waves, and see what you get.

6. ### WannabeNewton

5,859
Yes electric and magnetic fields can be out of phase e.g. if you consider certain superpositions of plane waves of certain polarizations: you can easily construct standing waves from superpositions of two oppositely propagating circularly polarized plane waves so as to have the electric and magnetic fields ##\pi/2## radians out of phase in time or space.

7. ### ANvH

54
See my previous reply: If using the plane wave then the math tells you that the ##E## and ##B## fields are in phase. This is what DaleSpam (https://www.physicsforums.com/newreply.php?do=newreply&noquote=1&p=2892896) on page 2 of that thread also showed.

In my original post I am contesting the notion, the physical interpretation, the interpretation of a rotational field, DaleSpam gave in a previous thread.

Again, is there a merit in saying that when the curl of a field is zero, that then the field it self is zero? If there is merit, I will try to come up with something that would make sense. Right now I do not see the point when I am wrong in how to interpret the curl of a field.

### Staff: Mentor

This is simply incorrect. While it is true that a zero magnetic field has zero curl, the reverse is not always true (i.e. zero curl does not imply zero magnetic field).

See here for an example of non-zero fields with zero curl.
http://en.wikipedia.org/wiki/Conservative_vector_field#Irrotational_vector_fields

Note, a plane wave is not irrotational everywhere, but where dE/dt=0 it is easy to show that ∇xB=0 also.

The first part is correct, but the second part is wrong. The curl of a plane wave is maximal when the field is zero.

I think that is the key, you are not correctly interpreting the curl of a vector field.

### Staff: Mentor

No, there is no merit in saying that. The reverse is true, but not that.

### Staff: Mentor

If you start plugging in ##\vec B## and ##\vec E##, beware that the equation above is incorrect. It should be
$$\vec \nabla \times \vec B = \frac{1}{c^2} \frac{\partial \vec E}{\partial t}$$

11. ### ANvH

54
Thanks for this refreshing remark.

For traveling waves:
An electromagnetic wave can be planar and circular polarized. A planar wave (polarized) can be decomposed into two dichroic circular waves. A circular wave (left or right) can be decomposed into two planar waves orthogonal to each other with a phase difference of ##\frac{\pi}{2}##. Each such planar wave can again be decomposed into dichroic circular waves, etc.

Given the above, what will be the basic wave for a photon, circular, planar? Does a circular polarized wave has the the electric and magnetic fields in phase?