Discussion of the Question 06/02

ISOTROPIC RADIATOR

The question was:

Is it possible to build and antenna that will radiate coherent electromagnetic radiation with intensity independent of direction?




(8/02) Y. Kantor: We got simultaneously (2/8/2002) two almost identical remarks from Jared Kaplan from Stanford University, US (e-mail jaredk@stanford.edu), and from Ben Zickel from Rafael, Israel (e-mail benz@rafael.co.il). Both comments point out that according to a topological theorem, frequently called as "the hairy ball theorem" it is impossible to cover the surface of a sphere with vectors of fixed size but continuously varying direction. At some point, such coverage must have discontinuity, i.e. jump in the direction of the vector field.

So, if we would attempt to create an antenna that produces radiation such that on a remote sphere we have fields oscillating with the same phase, this would violate the above theorem, since configuration of the fields at the moment of their (simultaneous) maximum would violate the theorem. Notice, however, that the problem required coherency, which implies fixed phase differences between various parts of the field, but not identical phases.
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