Find the power radiated using the Poyting vector

[Antiphon]

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.

 

[blueyellow]

Is z-hat in spherical coordinates just r-hat*cosθ?

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

?

 

[blueyellow]

So if

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

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

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

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

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

And

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

so

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

This means that E=-grad $\phi$-dA/dt=0

This can't be right, please help.

 

[blueyellow]

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.

 

[blueyellow]

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.

 

[vela]

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.