If E and H are static, phase has no meaning. If it is an antenna, E and H will not be static because the antenna will be operating at some frequency. Now assuming the antenna is operating at some frequency, the far field is essentially just a plane wave in free space. All of the conclusions about E and H in the far field come from theory of waves instead of any sort of antenna specific theory. Because free space is isotropic and lossess, the E and H fields are perpendicular to each other and are in phase. If the wave is in a lossy medium, E and H will be out of phase, although usually the phase difference is very small. If the medium is anisotropic, E and H will not necessarily be perpendicular to each other, although they likely be very close to being perpendicular.

The E and H field are in-phase in time in the Far Field and is the radiated field whereas the E and H field are always in a 90 degree spatial relationship to each other.

If E and H are static, phase has no meaning. If it is an antenna, E and H will not be static because the antenna will be operating at some frequency. Now assuming the antenna is operating at some frequency, the far field is essentially just a plane wave in free space. All of the conclusions about E and H in the far field come from theory of waves instead of any sort of antenna specific theory. Because free space is isotropic and lossess, the E and H fields are perpendicular to each other and are in phase. If the wave is in a lossy medium, E and H will be out of phase, although usually the phase difference is very small. If the medium is anisotropic, E and H will not necessarily be perpendicular to each other, although they likely be very close to being perpendicular.

This pdf gives the field equations

http://www.physics.princeton.edu/~mcdonald/examples/nearzone.pdf

In the near-field, an electric dipole has a large reactive fields that are similar to those you would expect from a capacitor and an inductor. These are 90 degrees out of phase, and are to do with the energy that has to be stored in the dipole for it to work properly (unless it is fed with a very low impedance the initial current is too low to radiate away the power being delivered, and the power builds up in the resonant circuit until the current is high enough to radiate away the power coming in).

There are also small (compared to the reactive fields) parts of the electric and magnetic fields that are in phase, and these carry power away. The out of phase fields drop off as 1/r2 and 1/r3 but the in-phase fields drop off as 1/r, so that although the reactive (out of phase) fields are much larger close in, the radiating (in-phase) fields dominate in the far-field, where the reactive fields die away completely for all practical purposes.

The equations are the result of applying the Biot-Savart laws to the curents in a short dipole.

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_chap10.pdf

The pdfs are not my work but say it much better than I can.

There is no such thing as DC - if it was switched on last week it has frequencies in the region of 1 per day to one per month. Very low but not DC. However, the wavelength for these frequencies is millions of kilometers so all measurements are in the very very near field, so you never see the in-phase components which are miniscule compared to the reactive fields.

Dear Colleague Muhammad,

welcome,

The far fields, the electric and the magnetic field form the electromagnetic wave which is dynamic travelling with speed of light in space. The two fields are coupled with each other when one grows the other grows and when it is attenuated the other is attenuated. In fact the magnetic field represents the kinetic energy in the wave while the electric field represents the potential energy in the wave. During the propagation, one form is transferred to the other alternately. If you annihilate one form the other will be be also finished.

Mathematically we express the fields of the electromagnetic wave in free space as:http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/emwv.html

Ex= Em cos ( wt - 2pi z/ lambda),

Hy= Hm cos (wt - 2pi z/ lambda)

Em/Hm = Zw where Zw is the wave impedance.

w is the frequency, lambda is the wavelength of the wave,

According to this formulation E and H are perpendicular to each other in space. They have the same phase in time wt.

If the electromagnetic wave is propagating in a lossy medium, the wave impedance will be complex, this means that there will be some form of phase shift between E and H.

Best wishes

What about E and H in Nearfield?

The near field is a localized field associated with its originating voltage or current. The origin of the electric field may be electric charges or a voltage gradient. It can be originated by time varying magnetic field. The capacitor model can be used to understand the electric field.

.The magnetic field can be originated by an electric current. It can also be originated by time varying electric field. The inductor can be used to understand the origin of the magnetic field.

The antenna may take a form of a capacitor , an inductor or a form of dipole. So the near field may be dominantly electric field or magnetic field. OR a combination of an in dependent magnetic and electric field in case of a dipole. The important thing is to radiate these fields must be time varying. So, in generalو near electric field and magnetic field are not related such as those of the far field.

Best wishes

Thanks.