I performed a simulation using HFSS for rectangular waveguide and here is the vector plot of E and H fields. 

I believe that the E and H fields are in phase with each other.

But why are E fields not in loops as explained by Guass's Law for EM waves (for JC = ρV =0). The E fields are sinusoidal in nature and H fields are in loops.

      In any case, the electromagnetic waves have to have E and H orthogonal fields,. If you do not get it making a simulation, then it means that something is wrong there.

In phase dear Suhas

Yes sir, Daniel Baldomir; they (E and H fields) are perpendicular to each other. My doubt is why E fields don't form loops? Why sinusoidal?  

Thank you Ariadne Tsambani madam! I see they are in phase with each other and perpendicular to each other and to the direction of propagation in the .pdf file you have attached!

     For applying Gauss law and enclosing electric charges (I suppose you refer to this), you have to have static charges. But when an electromagnetic field is on you have both fields and you do not obtain a simple closed loop, the form depends strongly of the frequency and the boundaries.

Dear Sushas.

In spite that the volume density of charge is zero, there is surface charge on the walls of the metal waveguide. These surface charges are the scalar sources of the electric field. Thus the electric field IS NOT solenoidal at all. However, the magnetic field IS SOLENOIDAL (no scalar sources). Thus, while the magnetic field lines do not start and finish in some specific place (this might means that the magnetic field lines are closed, although this is not the only possibility), the electric field lines for the fundamental TE10 mode in the rectangular waveguide go from the bottom wall to the top wall of the rectangular waveguide (and viceversa). In these walss there are surface charges accounting for this possibility.

Doesn't a material's volume include its surface? If not, don't we need separate Gauss's Laws for surface charge densities and volume charge densities?

If I have properly understood your problem, you have made simulations for a closed metallic waveguide. The walls of the waveguide are treated as the boundary conditions of the problem. Over the surface of the metal (or perfect electric conductor) walls you will have surface charge. The electric field vector is perpendicular to those walls and its value on the boundary is proportional to the surface charge density. Thus, the problem in the air region within the waveguide has not free volume charges, but the boundaries do support surface charges, in such a way that the electric field has scalar sources. This is not the case for the magnetic field.

Thank you sir Francisco Medina for your reply!

Dear Mr. Suhas Dinesh,

Since you were interested on phase lag between E and H, note that the TE10 mode of the waveguide has a z-component of the magnetic field. This component is 90 degrees out of phase with respect to the y-component of the electric field  (the only existing component for this field). This explains that transverse Poynting vector is purely imaginary, as required by the fact that the fields along the y-direction correspond to  a perfect standing wave. The transverse components (to z) of E and H are in phase thus leading to real Poynting vector, associated with real power flux along the z-component. 

Best regards,

Francisco

  Dear Suhas, 

  I have thought that your question was too general and basic, but I see that Prof.Medina was chosen, very cleverly, one TE10 mode for answering you.

   In this case the magnetic field H goes in the propagation direction, while the electric field E is orthogonal, giving rise  to loops of electrons in the faces. But if you have TM11 you do not obtain such a loops at all assuming a perfect conductor (no real metal with finite conductivity and eddy currents).

   In any case:

   1. Gauss law is not going to give you any information of the charges because it must give you always zero, because the you have a neutral perfect conductor.

   2. You need to take care in your simulations of the commensurability between the lenght wave and the boundary conditions.

   3. You are working with pure TE modes, which are the only ones to provide you with perfect loops on the faces.

Dear Suhas,

In brief, a linear polarized electromagnetic wave has a zero phase between B and E vectors, while for a circular polarized waves there are 90 degrees between B and E. More, for elliptically polarized waves the phase between B and E can be between 0 and 90 degrees (> 0 and< 90).

Best regards,

Marian

  It may seems that you can got waves where E and B are not ortogonal, after the post of Marian. This is false! These fields are always orthogonal.

   Linear polarization stands for waves where the electric field E or the magnetic B are constrained to be along the propagation plane. The phase is zero for the oscillations or , in other words, they are in phase. But the fields are always orthogonal and with 90 degrees between them.

Sir Daniel Baldomir: I understand that E and H fields are always perpendicular to each other and perpendicular to the direction of propagation.This is obvious from Maxwell's laws.

I hope you are trying to point out orthogonality (perpendicular with no time phase shift) and 900 phase shifts (not perpendicular, like I and V in a capacitor; I=C*dV/dt, V=sinwt then I=wcoswt).

Sir Marian Toma: Isn't Ex and Ey for circularly polarized waves 900 out of phase with Ex = Ey, and for elliptically polarized waves being Ex not equal to Ey with phase shift of 00 to 900 between these vectors?

Am I missing any other details for circular and elliptically polarized waves?

    Dear Suhas,

    The circular and eleptic polarizations corresponds to plane waves with special amplitudes relations. The most general form to see the different possibilities of polarization is to go to Poincare sphere as it is made in Optics.

   If the electromagnetic wave is composed of two plane waves of equal amplitude by differing in phase by 90°, then the light is said to be circularly polarized. MeanwhiIe if two plane waves, of differing amplitude, are related in phase by 90°, or if the relative phase is other than 90° then the light is said to be elliptically polarized.

Dear Sir,

Does two plane waves have to be both E fields? I am asking because of sir Marian Toma's reply above regarding polarization!

"...for a circular polarized waves there are 90 degrees between B and E. More, for elliptically polarized waves the phase between B and E can be between 0 and 90 degrees (> 0 and< 90)."

Thank you for your reply sir!

   Don´t worry Suhas,

   It is very simple. If the oscillation of the fields is in one fixed plane, then you have linear oscillation, and if the electric field rotates (the same must happen with the magnetic) without changing the strength then we have circular polarization and if the strength changes you have elliptic one (which generalized the circular one).

   The intermedia cases are represented with Stokes vectors in the Poincare sphere that you can see in any text book of Optics.

E and M fields in vacuum are always in phase! There is no room for discussing opinions.

Dear Suhas,

You may not have noticed this, but in Maxwell's theory, if you set both fields 180 degrees out of phase, you end up with exactly the same configuration as when both fields are in phase.

However, when dephased by 180 degrees, the very foundation of Maxwell's theory to the effect that both fields mutually induce each other can be implemented for localized self-sustaining quanta while maintaining the triple electromagnetic orthogonal relation, as de Broglie hypothesized in the 1930's.

If interested, you will find the details in this recent paper:

https://www.omicsonline.org/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.pdf