Electromagnetic wave and the phase between the E and B fields

In summary: There is some merit to saying that when the curl of a field is zero, that then the field it self is zero. However, this is only true if you are using a valid solution to the equations. If you are using a plane wave as your solution then the E and B fields are in phase. If you are using a spherical wave then the E and B fields may be out of phase.
  • #1
ANvH
54
0
https://www.physicsforums.com/showthread.php?p=2881300#post2881300

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:

Wannabeagenius is correct. They are in-phase, not 90 degrees out of phase.

If you look at Maxwell's laws in vacuum you will find that it is not quite corect that "a changing magnetic field induces an electric field". It is more correct to say "a changing magnetic field induces curl of an electric field" or in other words "a changing magnetic field (in time) induces a spatially changing electric field". When you express it correctly you immediately see that the electric and magnetic fields should be in phase.

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.
 
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  • #2
For a correct analysis, please review http://www.phy.iitb.ac.in/~dkg/PH-102/emw.pdf [Broken]

The E and B fields are in phase.

This is often used as a homework problem in your basic electromagnetic field theory course.
 
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  • #3
UltrafastPED said:
For a correct analysis, please review http://www.phy.iitb.ac.in/~dkg/PH-102/emw.pdf [Broken]

The E and B fields are in phase.

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

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.
 
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  • #4
ANvH said:
therefore
##E=E_{0}sin(\omega t - kx)##​
is incompatible with the Maxwell equations.

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
ANvH said:
Sorry, but I contest the planar wave as a solution.

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
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
Nugatory said:
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.

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 [Broken]) 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.
 
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  • #8
ANvH said:
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.
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.

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

ANvH said:
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.
I think that is the key, you are not correctly interpreting the curl of a vector field.
 
  • #9
ANvH said:
Again, is there a merit in saying that when the curl of a field is zero, that then the field it self is zero?
No, there is no merit in saying that. The reverse is true, but not that.
 
  • #10
ANvH said:
##∇ \times B=\frac{∂E}{∂t}##​

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

Thanks for this refreshing remark.

WannabeNewton said:
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.

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?
 

1. What is the relationship between the E and B fields in an electromagnetic wave?

In an electromagnetic wave, the electric field (E) and magnetic field (B) are perpendicular to each other and both are perpendicular to the direction of the wave's propagation.

2. What is the phase difference between the E and B fields in an electromagnetic wave?

The phase difference between the E and B fields in an electromagnetic wave is 90 degrees. This means that when one field is at its maximum value, the other field is at its minimum value, and vice versa.

3. How does the phase difference affect the properties of an electromagnetic wave?

The phase difference between the E and B fields is responsible for the oscillating nature of an electromagnetic wave. It also determines the polarization of the wave, which is the orientation of the electric field.

4. Can the phase difference between the E and B fields change?

Yes, the phase difference between the E and B fields can change depending on the medium through which the wave is traveling. For example, in a denser medium, the phase difference may decrease, resulting in a change in the wave's speed.

5. How is the phase difference between the E and B fields measured?

The phase difference between the E and B fields can be measured using specialized instruments such as a wave meter or an oscilloscope. These instruments can detect and compare the amplitudes and frequencies of the two fields to determine the phase difference.

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