# Poynting vector: Why We-Wm

• mohammed.omar
In summary, the conversation discusses the interpretation of the Poynting theory and why the terms for stored electric and magnetic field are subtracted from each other instead of added together. It is explained that this is due to the lagrangian and the dynamics of the objects, which is similar to the lagrangian for electromagnetic interaction. The concept is further clarified by comparing the phasor representation and time-domain representation of the fields. The conversation also recommends a textbook for further understanding of the topic.

#### 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?

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.

Hi kcdodd,

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

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).

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 :)

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

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?

mohammed.omar said:
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.

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

## 1. What is the Poynting vector?

The Poynting vector is a mathematical concept in electromagnetism that represents the direction and magnitude of energy flow in an electromagnetic field. It is named after physicist John Henry Poynting.

## 2. How is the Poynting vector calculated?

The Poynting vector is calculated by taking the cross product of the electric field vector and the magnetic field vector at a specific point in space. It is represented by the symbol S and has units of watts per square meter (W/m²).

## 3. What is the significance of the Poynting vector?

The Poynting vector is significant because it represents the flow of energy in an electromagnetic field. It is used to calculate the intensity of electromagnetic radiation and is essential in understanding the behavior of electromagnetic waves.

## 4. How does the Poynting vector relate to the wave nature of light?

The Poynting vector is directly related to the wave nature of light. It shows the direction and magnitude of energy flow in an electromagnetic wave, which is essential in understanding the behavior and properties of light.

## 5. What is the difference between the Poynting vector and power density?

The Poynting vector represents the flow of energy in an electromagnetic field, while power density represents the amount of power per unit volume. The Poynting vector takes into account both the electric and magnetic fields, while power density only considers the electric field. Additionally, the Poynting vector is a vector quantity, while power density is a scalar quantity.