= Radiation cuts and receiving polarization
The outgoing radiation field has magnitude $|E_\theta|=|\ddot p(t_r)|\sin\theta/(4\pi\epsilon_0c^2r)$ and $B_\phi=E_\theta/c$. The radial <Poynting vector>, averaged over one cycle, gives
$$
\boxed{\frac{d\langle P\rangle}{d\Omega}=r^2\langle S_r\rangle
=\frac{\omega^4p_0^2}{32\pi^2\epsilon_0c^3}\sin^2\theta.}
$$
Thus the $(x,z)$ cut has a two-lobed $\sin^2\theta$ shape, with zeros along $\pm z$ and maxima along $\pm x$. In the $(x,y)$ plane, $\theta=\pi/2$ throughout and the cut is a circle: there is no azimuthal dependence. At fixed direction the flux falls as $1/r^2$, while <power> per solid angle is independent of observation radius.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2018-b3-dipole-pattern-cuts.png]
{title=Original angular-power cuts of a z-directed Hertzian dipole: two lobes in xz and a circle in xy}
As a receiver for propagation along $x$, the antenna responds to the incident <electric field> component along $z$. For linear polarization at angle $\psi$ to $z$, the available load <power> is proportional to $\cos^2\psi$: it is maximal for $z$ polarization and zero for $y$ polarization. Equal incident intensity in circular $yz$ polarization gives half the matched-linear <power>. This polarization response follows from the induced voltage's projection along the dipole axis.
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