If we designate the origin (the reference point from which all displacement vectors are measured) $\vec{0}$, and If we consider a sphere $\mathbb{B}\left(\vec{0},\mathcal{R}\right)$ of radius $\mathcal{R}$ and centered at $\vec{0}$, and say that outside this sphere the charge density $\rho$ and current density $\vec{J}$ are zero at all points and all instants, then it can easily be shown pretty directly using Jefimenko's Equations that for points where $r \gg \mathcal{R}$, if we define: $$ \vec{\mathcal{Q}}\left(\hat{r}, t\right) = \iiint_{\mathbb{B}\left(\vec{0},\mathcal{R}\right)} \frac {\partial} {\partial t} \vec{J} \left(\vec{s}, t+\frac {\vec{s}\cdot\hat{r}} c\right)\space dV\left(\vec{s}\right) $$ Then we have $$ \vec{E}\left(\vec{r},t\right) \approx \frac {\mu_0}{4\pi r}\left(\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}\right)\times\hat{r} $$ $$ \vec{B}\left(\vec{r},t\right) \approx \frac {\mu_0}{4\pi rc}\left(\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}\right) $$ $$ \vec{S}\left(\vec{r},t\right) \approx \frac {\mu_0}{16\pi^2 r^2c}\left|\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}\right|^2 \hat{r} $$ Now we can apply the Hairy Ball Theorem to deduce that the vector $\vec{\mathcal{Q}}\left(\hat{r}, t-\frac{r}{c}\right)\times\hat{r}$ must be zero for at least one value or $\hat{r}$, which as we know rules out the isotropic antenna.

Now my question is, since it is allowable to have exactly one direction $\hat{r}$ in which this vector (and consequently, the radiated power) is zero, is there any such antenna--one that exhibits a non-zero radiated power in all directions $\left(\theta,\phi\right)$ except one? Somewhat like the pattern exhibited by a Cardioid microphone -- a cardioid revolved around its axis?

  • $\begingroup$ Note, that EM radiation have polarization, so it is possible to produce radiation in all directions from one emitter. $\endgroup$ – user23660 Nov 12 '13 at 7:12
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    $\begingroup$ @user23660 did you mean it is possible or isn't possible? If you mean isn't, my question is -- is it essential for radiation emitted from an antenna in all directions to have identical polarizations? $\endgroup$ – Avijit Nov 12 '13 at 8:42
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    $\begingroup$ Are you interested in a null in one direction across all frequencies and for a physically realizable "antenna"? Or would a specific current distribution inside a ball at a single frequency with this property satisfy your question? What characterizes a "null", exactly zero power? Because what we often call nulls in patterns are many tens of dB down from the maximum, but they are often not zero. If you clarify some of these points, I think I have an answer for you. $\endgroup$ – rajb245 Dec 29 '13 at 17:04
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    $\begingroup$ I was looking for exactly zero power (not merely very low power). Even a single frequency will do. $\endgroup$ – Avijit Dec 29 '13 at 17:56
  • $\begingroup$ Why SHOULDN'T the inverse direction ( -r ) exhibit a vector = zero (therefore zero radiated power)? You are after all playing with orthogonality... $\endgroup$ – Noldor130884 Mar 17 '15 at 13:54

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