I wonder if one could experimentally detect electromagnetic radiation emitted from the charged sphere, if it spins with w = 30 Hz - 30 kHz frequency, hence, emit in the low radio diapason and diameter of the sphere approach micrometer range (let’s say r = 25 um)?
From classics electrodynamics one can estimate mean intensity emitted by accelerated charges from the surface of the sphere. Which is, from electric dipole moment, I = (2/3)q2r2w2/c3, where r – radius of the sphere, w – frequency of rotation, q – electric charge and c – speed of light. Or I = (q2w2/600c)(qrB/mc2)4, from magnetic momentum of the sphere, if external magnetic field B is applied and m – is mass of the sphere.
Certainly, presence of magnetic field makes situation better. But anyway, what would it take to perform such detection in most simple way possible?
Probably put the rotating sphere in an appropriate optical cavity? Otherwise make a heterostructure with wave-guides at critical dipole radiation angles of the spinning sphere. I'm quite interested how you will charge and spin a sphere of that size though!
Dear Maksym,
Don't be worry with the form of measuring the electromagnetic radiation of your sphere. The amount of radiation is zero !!!
Dear Daniel,
Could you explain why it is zero? Or do you consider field strength of the order of ~10^-40 to be zero?
Strong external magnetic field, in principle, should rise the intensity of radiation to acceptable level.
Dear Marksym,
If you have one only body with spherical symmetry it is easy to prove that the Larmor radiation formula gives you exactly zero radiation. I am not speaking of low value but zero. It is true that in the practice you cannot get a pure sphere and perhaps you get a non spherical symmetry which can provide you with a very small value.
Dear Maksym,
I guess while part of the problem is measuring a very low quantity, the even more difficult part is to do it with sufficient accuracy because you have to measure in the near field (wavelength between 104 to 107 m !) where the reactive components with 1/r2 and 1/r3 dependency are considerably stronger than the radiation you are interested in.
BTW, could you please point me to the source of the formula for the magnetic dipole? A quick search did not provide any treatment taking the static B field into account. Thanks in advance!
Dear Daniel,
if I'm not mistaken the Larmor radiation is indeed zero in this case but since the spin of the sphere oscillates it can be modeled by an oscillating magnetic dipole; so we can expect dipole radiation instead.
Dear Frederik,
Thanks for your input. The sphere could be levitated and spun by a laser beam. The charge could be transferred to e.g. a liquid droplet by applying a voltage to the nozzle of a droplet generator. If you'd like to, I can share an article with detailed description on how it is done.
Dear Joerg,
The formula is given in a "Collection of problems on classical electrodynamics" by Alekseyev, problem #318, on page 98 (see attached file). There it is asked to calculate intensity of radiation from a homogeneous, charged sphere. In the end of the book there is an answer to this problem (page 225). Also, I found a "tutorial" with detailed solution to this problem on page 12 (see second attached file).
Unfortunately, both files are in Russian. But I hope you can make sense out of formulas.
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