Calculate Poynting Vector for Given Fields

In summary, the conversation discusses calculating the pointing vector for the magnetic and electric fields, and finding the real parts of these fields. The real part of B is calculated assuming a particular direction for P_omega, while the real part of E is more complex due to the combination of imaginary and real terms. The possibility of using imaginary numbers and phases to simplify the calculations is also mentioned.
  • #1
LocationX
147
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I am asked to calculate the pointing vector for the following fields:

[tex]\vec{B}=k^2 \frac{e^{ikr}}{r} \left( 1+\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}}[/tex]

[tex]\vec{E}=\frac{i}{k} (\vec{\nabla} e^{ikr}) \times \left( \frac{k^2}{r} \left(1+\frac{i}{kr} \left) \hat{r} \times \vec{p_{\omega}} \left) + \frac{i}{k} e^{ikr} \vec{\nabla} \times \left( \frac{k^2}{r} \left( i +\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}} \right) [/tex]

We know that:

[tex]\vec{S} = \frac{c}{4 \pi} Re(\vec{E}) \times Re(\vec{B}) [/tex]

We know that:

I can figure out [tex]Re(\vec{B})[/tex] assuming that P_omega points in the z direction:

[tex]Re(\vec{B})=k^2 p_{\omega} \frac{e^{ikr}}{r} sin \theta \hat{\phi} [/tex]

since the imaginary term in B vanishes when taking the real part.

I am not sure how to calculate the real part of E, any thoughts would be appreciated.
 
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  • #2
Why do you want to assume a particular direction for P_omega? If I remember correctly, if you play your cross-products right, you should get an expected result ...

You're forgetting the imaginary part of e^ikr. And this imaginary part will multiply the imaginary part of the other factor in B and result in another real contribution.
 
  • #3
turin said:
Why do you want to assume a particular direction for P_omega? If I remember correctly, if you play your cross-products right, you should get an expected result ...

You're forgetting the imaginary part of e^ikr. And this imaginary part will multiply the imaginary part of the other factor in B and result in another real contribution.

we assume a particular direction for P_omega so that r x p_omega will give the sin(theta) term

I am having trouble with finding the real part of E because I'm not sure how to find the real parts when imaginary terms are being crossed with real terms, any ideas?
 
  • #4
LocationX said:
... I'm not sure how to find the real parts when imaginary terms are being crossed with real terms, any ideas?
Re x Re = Re.
Im x I am = (-)Re.
Re x I am = Im.
Im x Re = Im.

You may also use i = e^ipi/2, and add phases to keep the expressions in polar form. In principle, both of these should be possible; however, choosing which way is more convenient comes with experience. Try both, and you will start to develop an intuition for it.

EDIT: Oh, wait, your expression for S is different than what I'm used to. I use Re(ExB*), or actually Re(ExH*). Sorry for the confusion. Anyway, you can't have Re(something) = something x e^ikr.
 
Last edited:

1. What is the Poynting vector?

The Poynting vector is a mathematical quantity that describes the direction and magnitude of electromagnetic energy flow in a given region of space. It is represented by the symbol S and is measured in watts per square meter.

2. How is the Poynting vector calculated?

The Poynting vector is calculated by taking the cross product of the electric field vector (E) and the magnetic field vector (H) at a specific point in space. The formula is S = E x H, where x represents the cross product operation.

3. What do the components of the Poynting vector represent?

The x and y components of the Poynting vector represent the direction of energy flow, while the z component represents the magnitude of energy flow. This means that the Poynting vector can point in any direction and its magnitude can vary depending on the strength of the electric and magnetic fields.

4. How is the Poynting vector used in electromagnetism?

The Poynting vector is an important tool in understanding and analyzing electromagnetic phenomena. It is used to calculate the amount of energy being carried by electromagnetic waves, such as light, and to determine the direction in which this energy is propagating.

5. Can the Poynting vector ever be negative?

Yes, the Poynting vector can have a negative value. This usually occurs when the electric and magnetic fields are out of phase with each other, resulting in a cancellation of energy flow. In this case, the Poynting vector points in the opposite direction of energy propagation.

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