The electric field of synchrotron radiation
A point charge moving uniformly around a circle produces an electric field pattern which co-rotates with it, constituting, for relativistic motion, synchrotron radiation. Surprisingly perhaps, the wealth of knowledge on synchrotron radiation does not seem to include explicit knowledge of the field itself, and of the consequent field lines. As with any relativistic motion, there is an obstruction to writing an explicit formula for the field; evaluation of the retarded time requires solving an implicit equation. However, as the relativistic limit is approached the field grows very strong in a very confined ribbon region shaped like a spiral watch spring. Here, the field can be written as an explicit scaling, or universal similarity, formula, which is our main result. From it the field lines can be derived analytically. In terms of scaled coordinates along the directions of the length, width and thickness of the ribbon, they twist in two side-by-side bundles in ‘bipolar’ cylinder surfaces, mirror symmetric about the orbit plane.