Physical interpretation of j in E&M field equations

[FrankJ777]

Hi all. I've been trying to study microwave and electromagnetic engineering . I'm not sure how I should interpret j in some of the field equations. For example, for the field equations for a rectangular waveguide resonant cavity are:

E_{y} = E_{0} sin\frac{\pi x }{a} sin \frac{l \pi z}{a}
H_{x} = \frac{-j E_{0}}{Z_{TE}} sin\frac{\pi x}{a} cos \frac{l \pi z}{d}
H_{z} = \frac{j \pi E_{0}}{k \eta a} cos\frac{\pi x}{a} sin \frac{l \pi z}{d}

What is the physical interpretation of j and -j in the H field in the x z direction? Does that indication that they are 90° out of phase of the E field? Does it indicate phase in the sense of time or space? Or should i think of them as derivatives of phasors? I know that the fields are derived from the more general phasor form of Maxwell's equations:
∇ × E = - jωμH
∇ × H = jωεE
for which jω = \frac{\partial E}{ \partial t } and E is E_{0} e^{j \omega t}
which makes sense to me as I believe you can interpret jω as the sinusoidal frequency. But once the E and H fields have been derived as above it's no longer jω just j, so I've lost the sense of there meaning in the H fields. Could someone please explain how I should interpret them. Or anything else I seem to have screwed up in my thinking. Thanks a lot.

 

[dlgoff]

The vector potential is defined to be consistent with Ampere's Law and can be expressed in terms of either current i or current density j (the sources of magnetic field). In various texts this definition takes the forms

I got this from http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Regards

 

[FrankJ777]

Thanks, but I don't think that's correct in this instance. In my book current and current density are usually denoted at J₀ or I₀. I'm fairly certain that in this instance j is the same as the imaginary unit, i, indicating that it's coefficient is the imaginary part of a complex number. Also these field equations were originally derived from;
∇×E = -jωμH
∇×H = jωεE
so everything should be in terms of the E and H fields.

 

[dlgoff]

Thanks, but I don't think that's correct in this instance. In my book current and current density are usually denoted at J₀ or I₀. I'm fairly certain that in this instance j is the same as the imaginary unit, i, indicating that it's coefficient is the imaginary part of a complex number. Also these field equations were originally derived from;
∇×E = -jωμH
∇×H = jωεE
so everything should be in terms of the E and H fields.

Okay. j is imaginary. Let's try again.

https://docs.google.com/viewer?pid=bl&srcid=ADGEESi9UJuwjRl0PQZfZ1Y-asjRs3wl3kfKwoQ8mW7cOLofGKnQ_c7feBTa6eLOjz6zE9QU-tsSVK5ZJ-9Xe9Sq1ON02oQnr9mV3ZWsMoZ2_fuR04ei5lCQGosrIeVy1KdR1w5kctnT&q=cache%3A18T64B7N10EJ%3Awww-ece.rice.edu%2F~daniel%2F262%2Fpdf%2Flecture02.pdf%20&docid=871d606ee53357c95f417addf3c94103&a=bi&pagenumber=20&w=800

http://www-ece.rice.edu/~daniel/262/pdf/lecture02.pdf

 

[yungman]

Hi all. I've been trying to study microwave and electromagnetic engineering . I'm not sure how I should interpret j in some of the field equations. For example, for the field equations for a rectangular waveguide resonant cavity are:

E_{y} = E_{0} sin\frac{\pi x }{a} sin \frac{l \pi z}{a}
H_{x} = \frac{-j E_{0}}{Z_{TE}} sin\frac{\pi x}{a} cos \frac{l \pi z}{d}
H_{z} = \frac{j \pi E_{0}}{k \eta a} cos\frac{\pi x}{a} sin \frac{l \pi z}{d}

What is the physical interpretation of j and -j in the H field in the x z direction? Does that indication that they are 90° out of phase of the E field? Does it indicate phase in the sense of time or space? Or should i think of them as derivatives of phasors? I know that the fields are derived from the more general phasor form of Maxwell's equations:
∇ × E = - jωμH
∇ × H = jωεE
for which jω = \frac{\partial E}{ \partial t } and E is E_{0} e^{j \omega t}
which makes sense to me as I believe you can interpret jω as the sinusoidal frequency. But once the E and H fields have been derived as above it's no longer jω just j, so I've lost the sense of there meaning in the H fields. Could someone please explain how I should interpret them. Or anything else I seem to have screwed up in my thinking. Thanks a lot.

In complex number j=e^{j\frac{\pi}{2}}=1∠90^o\;\hbox { and } \; -j=e^{-j\frac{\pi}{2}}=1∠-90^o

But I think you should write the formulas in vector form, I am only guessing the direction of the vectors by the subscripts like Ex is in x direction. What you gave are only the scalar value of the vectors.

Regarding to the disappearance of the jωt, your equation is in phasor form where

\vec E= Re[\tilde E e^{j\omega t}] \;\hbox{ where } \;\tilde E=\hat y E_0e^{-j\beta R}

BTW,

\frac {\partial E}{\partial t}= j\omega E_0 e^{j\omega t} \;\hbox { not }\; j\omega

Where your Eo has to contain the phasor function.

 

[FrankJ777]

Thanks guys.
OK Yungman, so you think the j represents a 90° phase shift in the orientation of the coordinates of the H fields. That's kind of something I was thinking, but I can't figure out how a 90° shift like j puts you on the x axis, while a -90° shift from -j puts you on the z axis. Also i was thinking because the field were derived from;
∇×E = -jωμH
∇×H = jωεE
that they must be related to the frequency jω, although ω does not seem to be factored into the fields in this case.
Looking at what dlgoff posted seems kind of close, but that would seem to indicate that the H field are purely imaginary, and can not be measured. I don't think this is the case though.

I could post the full derivation from the textbook I'm referencing if that would help??

 

[yungman]

Thanks guys.
OK Yungman, so you think the j represents a 90° phase shift in the orientation of the coordinates of the H fields. That's kind of something I was thinking, but I can't figure out how a 90° shift like j puts you on the x axis, while a -90° shift from -j puts you on the z axis. Also i was thinking because the field were derived from;
∇×E = -jωμH
∇×H = jωεE
that they must be related to the frequency jω, although ω does not seem to be factored into the fields in this case.
Looking at what dlgoff posted seems kind of close, but that would seem to indicate that the H field are purely imaginary, and can not be measured. I don't think this is the case though.

I could post the full derivation from the textbook I'm referencing if that would help??

I change my original post, I don't think the j has anything to do with the direction of H. If you look at the phasor equation, j=e^{j\frac{\pi}{2}} is a phase shift between the E and H ALONG the direction of propagation. This has nothing to do with the direction of E in y, H in x and z.

Remember if

E= jE_0 e^{-j\beta R}= E_0e^{-j(\beta R - \frac {\pi}{2})}
E= -jE_0 e^{-j\beta R} = E_0e^{-j(\beta R + \frac {\pi}{2})}

Where R is at the direction of propagation. This only mean the peak of the E and H is off by 90 degree along the direction of propagation, NOT the angle between E and H.

Edited: equations has been modified.

 

[yungman]

I draw out a TEM wave propagates in z direction. j is just a simple phase shift of the H. It has nothing to do with H in y direction and E and H are in perpendicular direction to each other.

 

[FrankJ777]

Yeah, that interpretation seems to make more sense. What is also confusing to me though is I thought that in general the E and H fields were in phase unless in a medium with a complex impedance. These seem to show that in a vacuum that the E and H fields are 90° out of phase. I'll have to dig further in the book and do a little more research to see if i can figure out what's going on. Thanks a lot for the hard work on helping me with this!

 

[yungman]

Yeah, that interpretation seems to make more sense. What is also confusing to me though is I thought that in general the E and H fields were in phase unless in a medium with a complex impedance. These seem to show that in a vacuum that the E and H fields are 90° out of phase. I'll have to dig further in the book and do a little more research to see if i can figure out what's going on. Thanks a lot for the hard work on helping me with this!

Only when TEM wave travel in lossless medium, both are in phase. But yours is in cavity. I just go by straight interpretation of your equations.

In fact, you should leave the equation in exponential form to take into account of the j before translating to sin and cos function as j represents a phase shift of ∏/2 and is together with the βR. so you should get something like sin(\frac{πR}{a}+π/2)

From your equation:

E_{y} = E_{0} sin\frac{\pi x }{a} sin \frac{l \pi z}{a}
H_{x} = \frac{-j E_{0}}{Z_{TE}} sin\frac{\pi x}{a} cos \frac{l \pi z}{d}
H_{z} = \frac{j \pi E_{0}}{k \eta a} cos\frac{\pi x}{a} sin \frac{l \pi z}{d}

The direction of propagation is in xz plane as your propagation constant has \sin\frac{\pi x }{a} \sin \frac{l \pi z}{a}. Obviously it is TE wave as E is normal to the direction of propagation and not H.

 

[FrankJ777]

Yes. The equations are for fields in a cavity but do not specifiy if they are in a lossless medium or not. That would depend on β and Z₀ I think. Also these are TE waves, not TE waves.

 

[yungman]

Yes. The equations are for fields in a cavity but do not specifiy if they are in a lossless medium or not. That would depend on β and Z₀ I think. Also these are TE waves, not TE waves.

I just add the last part of into the last post. Yes, it is a TE wave from your formulas as the direction of propagation is in xz plane. As I said, I don't know TE wave, I have no comment on this. But medium being lossless is only one part of the big picture, there might be other factor that add a phase shift between the two waves. Case in point, yours is TE in cavity.