Find the power radiated using the Poyting vector

In summary: So if you're missing a word, it might be because I skipped over it.In summary, you would calculate the power radiated by a model for electric quadrupole radiation by integrating over the area of a closed surface, e.g. a sphere.
  • #36
The radiation zone is where the transverse field dominate over any radial components. It is defined to be R = 2D^2/lambda where D is the diameter of the source and lambda is the wavelength.
 
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  • #37
Is z-hat in spherical coordinates just r-hat*cosθ?

So A is now:
A=[itex]\frac{-\mu_{0}p_{0}\omega^{2}d}{4\pi cr}[/itex]cos[itex]^{2}[/itex]θcos([itex]\omega[/itex](t-r/c))r-hat

?
 
  • #38
So if

grad [itex]\phi[/itex]=(d[itex]\phi[/itex]/dr]r-hat +(1/r)(d[itex]\phi[/itex]/dθ)θ-hat

=[itex]\frac{-\mu_{0}p_{0}\omega^{2}d}{4\pi}[/itex]cos[itex]^{2}[/itex]θ([itex]\frac{-1}{r}[/itex][itex]\frac{-\omega}{c}[/itex]sin([itex]\omega[/itex](t-r/c))+[itex]\frac{1}{r^{2}}[/itex]cos([itex]\omega[/itex](t-r/c)))r-hat
+[itex]\frac{1}{r}[/itex]([itex]\frac{-\mu_{0}p_{0}\omega^{2}d}{4\pi r}[/itex]cos([itex]\omega[/itex](t-r/c))(-sin 2θ))θ-hat

and dropping the (1/(r^2)) terms for the radiation zone it is:

grad [itex]\phi[/itex]=(d[itex]\phi[/itex]/dr]r-hat +(1/r)(d[itex]\phi[/itex]/dθ)θ-hat

=[itex]\frac{-\mu_{0}p_{0}\omega^{2}d}{4\pi}[/itex]cos[itex]^{2}[/itex]θ([itex]\frac{-1}{r}[/itex][itex]\frac{-\omega}{c}[/itex]sin([itex]\omega[/itex](t-r/c))r-hat

And

A=[itex]\frac{\mu_{0}p_{0}\omega^{2}d}{4\pi cr}[/itex]cos[itex]^{2}[/itex]θcos([itex]\omega[/itex](t-r/c))r-hat

so

dA/dt=[itex]\frac{-\mu_{0}p_{0}\omega^{3}d}{4\pi cr}[/itex]cos[itex]^{2}[/itex]θsin([itex]\omega[/itex](t-r/c))r-hat

This means that E=-grad [itex]\phi[/itex]-dA/dt=0

This can't be right, please help.
 
  • #39
Antiphon said:
The radiation zone is where the transverse field dominate over any radial components. It is defined to be R = 2D^2/lambda where D is the diameter of the source and lambda is the wavelength.

Thanks, but how would I apply this to the question?

So for the question R=(2(d^2))/(2pi/omega).

Does this have any implications on how I should do the question/approximate?

Thanks if you reply.
 
  • #40
I just realized, I made a mistake while expressing z-hat in spherical coordinates.

https://www.physicsforums.com/showthread.php?t=126702 - this told me that to find out what z-hat is in spherical coordinates, one has to do grad(z)=grad(r cos theta)=z-hat.

Please tell me if what they say is wrong. I have found no other textbook or website that will tell me how to express z-hat in spherical coordinates.
 
  • #41
You could do it that way, but there are other ways. For example, you should know that because ##\hat{r}##, ##\hat{\theta}##, and ##\hat{\phi}## form an orthogonal basis, you have that ##\hat{z} = (\hat{z}\cdot\hat{r})\hat{r} + (\hat{z}\cdot\hat{\theta})\hat{\theta} + (\hat{z}\cdot\hat{\phi})\hat{\phi}##. Presumably, you have expressions for ##\hat{r}##, ##\hat{\theta}##, and ##\hat{\phi}## in terms of the Cartesian unit vectors, so that expression should be trivial to work out.

In any case, in 9.1.2, Griffiths mentions what ##\hat{z}## is in terms of the spherical unit vectors. Have you taken the time to read and understand what he did in that section? If you understand what he did there, this problem is very straightforward.
 
<h2>1. What is the Poyting vector?</h2><p>The Poyting vector is a mathematical concept used in electromagnetism to describe the direction and magnitude of energy flow in an electromagnetic field. It is represented by the symbol S and is defined as the cross product of the electric field and magnetic field vectors.</p><h2>2. How is the Poyting vector used to find the power radiated?</h2><p>The Poyting vector can be used to calculate the power radiated by an electromagnetic field by taking the dot product of the Poyting vector and the surface normal vector of the area being considered. This gives the power per unit area, which can then be integrated over the entire surface to find the total power radiated.</p><h2>3. What is the significance of finding the power radiated using the Poyting vector?</h2><p>Finding the power radiated using the Poyting vector is important in understanding the behavior of electromagnetic fields and their interactions with matter. It allows for the calculation of energy transfer and can be used to analyze the efficiency of various devices that use electromagnetic radiation, such as antennas and solar panels.</p><h2>4. Are there any limitations to using the Poyting vector to find power radiated?</h2><p>Yes, there are some limitations to using the Poyting vector to find power radiated. It assumes that the electromagnetic field is in a vacuum and does not take into account any losses due to absorption or scattering by materials. It also does not consider the effects of non-linear materials or time-varying fields.</p><h2>5. How is the Poyting vector related to the energy density of an electromagnetic field?</h2><p>The Poyting vector is directly proportional to the energy density of an electromagnetic field. This means that areas with a higher Poyting vector will have a higher energy density, indicating a higher amount of energy being transferred through that area. The Poyting vector can also be used to calculate the total energy stored in an electromagnetic field by integrating over the entire volume.</p>

1. What is the Poyting vector?

The Poyting vector is a mathematical concept used in electromagnetism to describe the direction and magnitude of energy flow in an electromagnetic field. It is represented by the symbol S and is defined as the cross product of the electric field and magnetic field vectors.

2. How is the Poyting vector used to find the power radiated?

The Poyting vector can be used to calculate the power radiated by an electromagnetic field by taking the dot product of the Poyting vector and the surface normal vector of the area being considered. This gives the power per unit area, which can then be integrated over the entire surface to find the total power radiated.

3. What is the significance of finding the power radiated using the Poyting vector?

Finding the power radiated using the Poyting vector is important in understanding the behavior of electromagnetic fields and their interactions with matter. It allows for the calculation of energy transfer and can be used to analyze the efficiency of various devices that use electromagnetic radiation, such as antennas and solar panels.

4. Are there any limitations to using the Poyting vector to find power radiated?

Yes, there are some limitations to using the Poyting vector to find power radiated. It assumes that the electromagnetic field is in a vacuum and does not take into account any losses due to absorption or scattering by materials. It also does not consider the effects of non-linear materials or time-varying fields.

5. How is the Poyting vector related to the energy density of an electromagnetic field?

The Poyting vector is directly proportional to the energy density of an electromagnetic field. This means that areas with a higher Poyting vector will have a higher energy density, indicating a higher amount of energy being transferred through that area. The Poyting vector can also be used to calculate the total energy stored in an electromagnetic field by integrating over the entire volume.

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