Solve EM Radiation Homework: Compute E & B with Monochromatic Point Source

[qoqosz]

Homework Statement

We are given monochromatic point source of EM radiation which power is P=100W. The task is to compute E(r) and B(r). We can assume that r is large enough to treat wave as a plane wave.

Homework Equations

The Attempt at a Solution

First of all - what for do we assume that for big r it is plane wave?
My solution to this task is:

In a sphere of radius r and thickness dr there is an amount of energy W: $$W = P dt = \frac{1}{\epsilon \mu} EB 4 \pi r^2 dr$$
Then $$P = \frac{1}{\epsilon \mu} EB 4 \pi r^2 c \iff EB = \ldots$$ and so on... I can easily calculate values of E and B but still - what for is the mentioned assumption?

 

[gabbagabbahey]

First, why is there a factor of $1/\epsilon$ in your expression? Do the units make sense?

If the fields aren't those of a plane wave, then the Poynting vector $\textbf{S}=\frac{1}{\mu_0}\textbf{E}\times\textbf{B}$ does not necessarily point in the radial direction and have magnitude $EB$. The fields themselves could also depend on the polar and azimuthal angles,

$$\implies P(r)=\oint\textbf{S}\cdot d\textbf{a}=\int_0^{\pi}\int_0^{2\pi}\textbf{S}\cdot\hat{\textbf{r}}r^2\sin\theta d\theta d\phi\neq\frac{4\pi r^2}{\mu_0}EB$$

in general.

 

[qoqosz]

Ok, thanks. I used $$\frac{1}{\mu \epsilon} EB$$ as an energy density not an energy flux.

 

[gabbagabbahey]

Ok, thanks. I used $$\frac{1}{\mu \epsilon} EB$$ as an energy density not an energy flux.

But that doesn't even have units of energy density...

 

[qoqosz]

You're right - I made stupid mistake :( Should be: $$\frac{1}{\mu c} EB$$