# How do you make a spherical radio wave?

A vertical rod, a usual dipole, produces radio waves in the horizontal plane, mostly in two opposite directions:

$\qquad \qquad \qquad \qquad \qquad$

If that is possible, how do you produce a spherical EM radiation? should the antenna be a (..n expanding and contracting) globe or a circle? How should the charges oscillate? and, lastly, would its energy decrease by 1/$4 \pi r^2$ and so its range would be rather short?

P.S. Someone said in a comment to
How is a spherical electromagnetic wave emitted from an antenna described in terms of photons?:

For some reason, my instinct is that a spherical electromagnetic wave cannot be emitted by an antenna. Instead, they can only be emitted by a charge. I guess that's cause I always think of an antenna as an object that has no net charge. – Carl Brannen

Is this true? can you explain how a charge, say an electron, can produce a spherical wave? Also, does the section (the area) of a charge carry any info about its force or anything else?

• The wave is actually coherent photons being emitting in every direction from the accelerated electrons. – Bill Alsept Apr 25 '16 at 14:46
• To your note: no, it is not true. – garyp Apr 25 '16 at 17:47
• If you're aware of the hairy ball theorem, it should be clear that a spherically symmetric vector wave has to be radially polarized. Gauss's law then prevents such a polarization in free space. – Emilio Pisanty Apr 25 '16 at 17:48
• Do you mean spherical polarized or circularly polarized? If circularly polarized, you can look though a data sheet for any circularly polarized FM broadcast antenna. shively.com/nav-fmantennas.php I might have completely missed the point of the question though. – user115455 Apr 25 '16 at 18:53
• Re: "Mostly in two opposite directions": No. The radiation pattern of a dipole is donut shaped. The emission from a vertically oriented dipole antenna is the same in every compass direction. It falls off with elevation (i.e., as the receiver goes above or below the central plane.) – Solomon Slow Apr 25 '16 at 19:02

A result known as Birkhoff's theorem forbids spherical electromagnetic radiation. The statement of the theorem is that any spherically symmetric vacuum solution to Maxwell's equations must be static. It is rather simple to prove. In a spherically symmetric solution $\mathbf E$ and $\mathbf B$ must be radial. Make an Ansatz, $$\mathbf E = E_0 \exp(i(\mathbf k\cdot\mathbf r-\omega t)) \hat r \quad \mathbf B = B_0 \exp(i(\mathbf k\cdot\mathbf r-\omega t)) \hat r$$ The wavevector $\mathbf k$ must be $\mathbf k = k\hat r$ for spherical symmetry. Now Ampere's law is $$\nabla\times \mathbf B = i\mathbf k \times \mathbf B = 0 = \partial_t \mathbf E = -i\omega \mathbf E$$ which implies $\omega = 0$, so that the field is static, or $E_0 = 0$. From Faraday's law $\nabla\times\mathbf E =- \partial_t \mathbf B$ you can see that if $E_0 = 0$ but $\omega \neq 0$, then also $B_0 = 0$.

The most general result for electromagnetic radiation is that in Coulomb gauge, in the radiation zone, the vector potential is $$\mathbf A(\mathbf x, t) = \frac{\mu_0}{4\pi }\frac{e^{i(kr-\omega t)}}{r} \int \mathbf J(\mathbf x') e^{-ik\hat{x} \cdot \mathbf x'} \, dx'$$ where $\mathbf J(\mathbf x')$ is the current in the source region, e.g., your antenna, and the current is assumed to have sinusoidal (harmonic) time dependence. [This is not a restriction because Maxwell's equations are linear and Fourier transform exists.]

The angular dependence is entirely in the integral over the source current. Thus to achieve some desired angular profile of the radiation, one needs to design $\mathbf J$ appropriately.

Your particular case of an oscillating sphere of charge actually does not radiate because it has only a monopole moment and there is no monopole radiation. A spheroidal charge distribution is treated by Jackson Classical Electrodynamics, Sec. 9.3. There Jackson shows that this arrangement leads to quadrupole radiation with a four-lobed distribution of radiated power. For a more in-depth discussion, read Ch. 9 in Jackson, which treats radiation in detail, including the angular distribution of radiated power from various sources.

• Does this imply that stars cannot emit equal light in all directions? – Octopus Apr 25 '16 at 18:49
• @Octopus I don't think so. Thermal radiation like from a star is inherently statistical. I think that provides a loophole big enough. – Robin Ekman Apr 25 '16 at 19:06
• @Octopus it is not coherently emitted in all directions. – Rob Jeffries Apr 25 '16 at 19:30

"If that is possible, how do you produce a spherical EM radiation?"

A spherically symmetric transverse field is topologically impossible - if it is required to be coherent and linearly polarized everywhere. This is the case for usual dipole or higher multipole radiation, as has already been pointed out in another answer.

On the other hand, an incoherent transverse field, otherwise known as an incoherent superposition of coherent fields, may well be spherically symmetric, because it no longer has a well-defined polarization. This is what we casually refer to as a "spherically symmetric e.m. radiation field". But this is boring.

The far more interesting stuff is that if we still require a coherent field, but

1. relax the requirement for uniformity in polarization,

or

1. make good use of already available metamaterials technology,

it is possible to put together something arbitrarily close to "spherically symmetric intensity". The corresponding devices are known as isotropic radiators (not to be confused with old omnidirectional antennas, which generally produce a doughnut-shaped field).

For alternative (1), example antenna arrangements that may generate virtually isotropic far-field intensity are given in arXiv:physics/0312023. They are variants of $\lambda / 4$ narrow U-shaped antennas or linear arrays of turnstile-antennas operated in phase. According to the paper, even for one pair of turnstile antennas the maximum intensity is only 1.08 times larger than minimum intensity. It is also pointed out that a spherical shell radiator may produce isotropic output from certain patterns of finite oscillating currents (see refs. therein).

Alternative (2) has been described in Phys. Rev. Lett. 111, 133901 (2013) (also available here). The idea is to reshape a dipole radiation pattern by means of suitably arranged metamaterial structures such that even the near-field becomes isotropic. Similar reshaping techniques are applicable in principle to any antenna output.

Bottom line is, looks like there are always possibilities... ;)

• @ udrv That's plenty interesting and it's worth expanding on that point (1) a bit more, but it's probably better to do that in a separate thread, which I would encourage you to answer (with some rep on that one too if you're interested). – Emilio Pisanty Feb 13 '17 at 12:54
• Thanks! Looking into it, but am still learning myself. In the meantime you'd probably be interested in this other paper of Matzner's I just found, hindawi.com/journals/ijap/2012/187123 (open access html version, but pdf download looks much better). It seems to provide the details, but have yet to go through it all properly. – udrv Feb 13 '17 at 16:14

One can use an easy to understand picture. Suppose, you have an antenna composed from 6 rods along the axes of a cartesian coordinate system. At one moment electrons together get accelerated outwards in all 6 rods. Using a hand rule one could see that the magnetic fields around the rods cancel each other out exactly.

So you are free to make an experiment with a hollow sphere, placing inside an electric source and an electronic device to move electrons outwards and inwards the sphere. You shouldn't be able to measure any radio wave.