# MTW Ex 21.23 Poynting Flux Vector 'out of the air'

• TerryW
In summary, the Lagrangian Density of the Hamilton-Jacobi functional, as described by MTW (21.100), is the appropriate functional for producing the required result. If it is not the appropriate functional, then an alternative should be used. The solution to the problem can be found by using this functional.
TerryW
Gold Member
Homework Statement
I'm just looking for a bit of guidance with this, to check if I am on the right track
Relevant Equations
MTW (21.100) See Below
##4\pi\mathcal L = -\mathcal e \frac{\partial A_i }{\partial t} - \phi\mathcal E^i{}_{,i} -\frac{1}{2}N\gamma^{\frac{1}{2}}g_{ij}(\mathcal E^i \mathcal E^j +\mathcal B^i\mathcal B^j) +N^i [ijk]\mathcal E^i\mathcal B^j## MTW (21.100)

I'm trying to produce the result required by the problem below by using the Lagrangian Density above as the Hamilton-Jacobi functional ##(S(g_{ij},A_m)## referred to below. So far, I haven't been able to see a way forward which might end up with the required result.

At this stage, I'm just asking if MTW (21.100) is the appropriate functional and if it isn't, what should I be using?

The answer to my question is yes. I found the key to the solution.

## 1. What is the Poynting Flux Vector?

The Poynting Flux Vector is a mathematical representation of the flow of electromagnetic energy through a given point in space. It is named after its discoverer, John Henry Poynting.

## 2. What does "out of the air" refer to in MTW Ex 21.23?

"Out of the air" refers to the fact that the Poynting Flux Vector is a measure of the electromagnetic energy flow through a point in space that is not associated with any particular material medium. It is often used to describe the flow of energy through empty space.

## 3. How is the Poynting Flux Vector calculated?

The Poynting Flux Vector is calculated by taking the cross product of the electric field vector and the magnetic field vector at a given point in space. This calculation takes into account the direction and magnitude of both fields to determine the flow of energy through that point.

## 4. What is the significance of the Poynting Flux Vector?

The Poynting Flux Vector is significant because it helps us understand and quantify the flow of electromagnetic energy through space. It is used in many areas of physics and engineering, such as in the study of electromagnetic radiation and the design of wireless communication systems.

## 5. How is the Poynting Flux Vector used in practical applications?

The Poynting Flux Vector is used in practical applications such as antenna design, electromagnetic wave propagation, and power transmission. It is also used in the study of electromagnetic radiation from sources such as the sun and other celestial bodies.

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