Why does the Poynting vector subtract stored electric and magnetic fields?

[mohammed.omar]

Hi All,

I have been revising the Poynting theory and I cannot interpret "physically" why do we have the terms of stored electric(We) and magnetic field (Wm) minus each other; doesn't it make more sense to have the total energy stored in the field added together when we talk about the Poynting theory as of a conservation law?

 

[kcdodd]

To get the total energy density you do add them together. You subtract them to get the dynamics, which is most likely rooted in the lagrangian of the same form.

 

[mohammed.omar]

Hi kcdodd,

Thanks for the prompt reply but I do not really get your point. Could you please elaborate more?

 

[kcdodd]

For Newtonian objects, the lagrangian is usually the kinetic energy minus the potential energy (T - V), as opposed to the total energy (T + V), which is a constant. But we use the lagrangian to determine the dynamics of the objects. The lagrangian for em interaction is a similar kind of idea. The dynamics are governed by (E^2 - B^2), instead of (E^2 + B^2).

 

[lavster]

in my notes it is added together. have a look at wikipedia too: http://en.wikipedia.org/wiki/Poynting's_theorem.

unless of course I am misunderstanding your question?

basically poyntings theorem tells you that the work done on the charges is equal to the decrease in energy stored in that field minus the energy that flowed out the surface (ie the energy transported by the fields.

I recommend the textbook "introduction to electrodynamics" by D. Griffiths if you haven't already looked at it - it is very good for most electrodynamic material for an undergraduate :)

 

[kcdodd]

There is a version for harmonic fields where you subtract the field densities, which seems counterintuitive at first.

 

[mohammed.omar]

Thanks lavster and kcdodd.

Griffiths book seems a good place to start :)

I'm using the harmonic field version, as kcdodd said, and I believe I'm starting to get grips of the "dynamic behavior" argument. It is a very neat explanation actually, and it has a very appealing agreement with circuit theory. Is there any reference that describes the problem or the technique as you describe it?

I've looked at Pozar's "Microwave Engineering" and Collin's "Foundations of Microwave Engineering". I noted that using the phasor representation, we use Wm-We, while using time-domain representation it has the form Wm+We. Does this mean that using the phasor representation we study the dynamic behavior while when using the more general time-domain representation we study the more general "large signal" case?

 

[Born2bwire]

Thanks lavster and kcdodd.

Griffiths book seems a good place to start :)

I'm using the harmonic field version, as kcdodd said, and I believe I'm starting to get grips of the "dynamic behavior" argument. It is a very neat explanation actually, and it has a very appealing agreement with circuit theory. Is there any reference that describes the problem or the technique as you describe it?

I've looked at Pozar's "Microwave Engineering" and Collin's "Foundations of Microwave Engineering". I noted that using the phasor representation, we use Wm-We, while using time-domain representation it has the form Wm+We. Does this mean that using the phasor representation we study the dynamic behavior while when using the more general time-domain representation we study the more general "large signal" case?

No, the two formulations are equivalent, the time-harmonic case is just a Fourier basis decomposition of the general time-domain fields. Remember though, to go back to the true time-domain you have to take the real part of the complex time-harmonic fields.

 

[mohammed.omar]

Tanks a lot. I believe I've got grips of it :).