Approaching the following problem:
A plane monochromatic electromagnetic wave with wavelength $\lambda = 2.0 cm$,
propagates through a vacuum. Its magnetic field is described by $
> \vec{B} = ( B_x \hat{i} + B_y \hat{j} ) \cos(kz + ωt) $,
where $B_x = 1.9 \times 10^{-6} T, B_y = 4.7 \times 10^{-6} T$, and $\hat i$ and $\hat j$ are
the unit vectors in the $+x$ and $+y$ directions, respectively.
What is $S_z$, the $z$-component of the Poynting vector at $(x = 0, y = 0, z = 0)$ at $t = 0$?
Question 1
What is the Poynting vector in a general sense? i.e. Abstractly, what am I trying to compute?
I know it is described as the directional energy flux density of an electromagnetic field. But, it is not traveling through a symmetric surface or anything of this nature so what am I trying to compute here?
Question 2
How do I actually compute the value I am looking for?
I know $\vec S \equiv \frac{\vec E \times \vec B}{ \mu_0}$ where $\vec E$ is the electric field, $\vec B$ is the magnetic field, and $\mu_0 = 4 \pi \times 10^{-7}$. But, I was under the impression that the magnetic field and the electric field are always perpendicular, why isn't $S_z = \frac{ c}{ \mu_0}$ since we are in a vacuum and $E = cB$?
Best Answer
The Poynting vector $\mathbf{S}$ represents the flow of energy in an EM field. Specifically, if $u$ is the energy density of the field, the Poynting vector satisfies the continuity equation for it: $$\frac{\partial u}{\partial t}+\nabla\cdot\mathbf{S}=0$$ in vacuum. (This is Poynting's theorem.)
In your particular problem, $E$ and $B$ are perpendicular and their cross product is proportional to the product of their amplitudes. Thus $$S_z={c\over\mu_0}B^2.$$ You then have to use your knowledge of $B$ to work out $S$.