Solving Poyntings Vector for a point charge

In summary, the Poynting vector, given by P = ExH = (mu0q2a2sin2(theta)/6pi2cr2)r, leads to the total power radiated, given by (mu0q2a2/6pi2c)\int(sin2(theta)/r2)(2pir2sin(theta)d\theta), over the whole volume. This is derived using Poynting's theorem and a spherical surface of radius r to represent the infinitesimal vector area element of the surface.
  • #1
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Homework Statement



It is found that Poyntings vector gives

P = ExH = (mu0q2a2sin2(theta)/6pi2cr2)r

This apparently leads to

Total Power = (mu0q2a2/6pi2c)[tex]\int[/tex](sin2(theta)/r2)(2pir2sin(theta)d[tex]\theta[/tex])

What I am unsure of is where the

(2pir2sin(theta)d[tex]\theta[/tex])

appears from. Can anyone help?
 
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  • #2
[strike]The power is over the whole of the volume, and not just one component of the volume.[/strike] When you are coming from Cartesian coordinates to http://en.wikipedia.org/wiki/Spherical_coordinate_system#Cartesian_coordinates", we find

[tex]
\int dV=\int_{-\infty}^\infty dx\int_{-\infty}^\infty dy\int_{-\infty}^\infty dz=\int_0^\infty r^2dr\int_0^\pi \sin\theta \,d\theta\int_0^{2\pi} d\phi
[/tex]
 
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  • #3
jdwood983 said:
The power is over the whole of the volume, and not just one component of the volume.

Errr... not quite...Poynting's theorem tells us that the power crossing a surface [itex]\mathcal{S}[/itex] is given by

[tex]P(\textbf{r})=\int_{\mathcal{S}}\textbf{S}\cdot d\textbf{a}[/tex]

(using [itex]\textbf{S}[/itex] to represent the Poynting vector and [itex]d\textbf{a}[/itex] to represent the infinitesimal vector area element of the surface)

To find the power radiated, one usually uses a spherical surface of radius [itex]r[/itex] (so that [itex]d\textbf{a}=r^2\sin\theta d\theta d\phi\mathbf{\hat{r}}[/tex])) and then takes the limit as [itex]r\to\infty[/itex] (since radiation escapes to infinity).
 

FAQ: Solving Poyntings Vector for a point charge

1. What is Poynting's vector for a point charge?

Poynting's vector is a mathematical concept used in electromagnetism to describe the flow of energy in an electromagnetic field. It is represented by the symbol S and is defined as the cross product of the electric field vector (E) and the magnetic field vector (B).

2. Why is it important to solve Poynting's vector for a point charge?

Solving Poynting's vector for a point charge allows us to calculate the energy flow and distribution in an electromagnetic field. This information is crucial in understanding the behavior of electromagnetic waves and their effects on objects and systems.

3. What is the formula for calculating Poynting's vector for a point charge?

The formula for calculating Poynting's vector for a point charge is S = E x B, where S is the Poynting vector, E is the electric field vector, and B is the magnetic field vector. This can also be expressed as S = (1/μ0) * (E x B), where μ0 is the permeability of free space.

4. How is Poynting's vector for a point charge related to the direction of energy flow?

Poynting's vector points in the direction of energy flow in an electromagnetic field. This means that the direction of the vector represents the direction in which energy is being transported by the electromagnetic wave. The magnitude of the vector represents the rate at which energy is being transported.

5. Can Poynting's vector for a point charge be used to calculate the energy density of an electromagnetic field?

Yes, Poynting's vector can also be used to calculate the energy density of an electromagnetic field. The energy density (u) is related to the magnitude of Poynting's vector by the equation u = |S|2 / c2, where c is the speed of light in a vacuum. This means that the energy density is directly proportional to the square of the magnitude of Poynting's vector.

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