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Here is a question that I've had for $\geq$ 30 years but only now am in a position to ask properly. Electromagnetic waves are "transverse" which according to the textbooks means that the amplitude (in this case E and B fields) are perpendicular to the direction of propagation.

Now, if we have a point source then the amplitude vectors should furnish a vector field defined on a small ball around that source but according to the aforementioned hairy-ball theorem such cannot exist in a continuous fashion. What is the way out of this?

Emilio Pisanty
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Martin Hofmann
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2 Answers2

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The answer is that physical sources generally do not emit isotropically; instead, the archetypical radiator emits in a dipole pattern with the emission concentrated along a plane, with two zeros in its intensity distribution.

That said, it is possible to produce so-called isotropic radiators.

  • One way is to have two orthogonal polarizations with complementary intensity distributions and then add them incoherently, i.e. having two sources without a definite phase relationship on the two, so they don't actually form a vector field. This is how light from a star, even when monochromatized, gets to have an isotropic intensity distribution.

  • The more interesting way is to do this coherently, by exploiting the fact that monochromatic EM radiation can quite happily accommodate circular polarizations. For more details, see How do coherent isotropic radiators evade the hairy-ball theorem?.

Emilio Pisanty
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The long distance component of electromagnetic radiation is dipolar, not monopolar. That means that you need to choose a direction for the dipole emitters before writing a generic wave solution. Only scalar fields can have monopolar terms, like pressure for example.

lurscher
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