Let us have some localized density of sources, S, in a plane, each of which produces a localized circular vector field. Let us work in polar coordinates. Let the density of sources, S = Aexp(-r^2/a^2) and let each source have circular vector field whose strength is given by exp(-(r-r_i)^2/b^2) and is in a clockwise circular direction about the point r_i where r_i is the location in the plane of the ith source.(adsbygoogle = window.adsbygoogle || []).push({});

The constants a and b are the length scales in this problem, a determines how far from the origin most of the sources will be found and b tells us how localized the field of each source is, a is much larger than b. The constant A is a number density such that the integral of S over the whole plane gives the number of sources which is a very large number.

I think I know qualitatively what the resultant vector field is like, far from the origin the field goes to zero and at the origin the field goes to zero, where the density S changes the greatest in the radial direction one will find the strongest resultant field, and the resultant field is circular about the origin. But it would be nice to prove this.

I am only looking for qualitative results. Any hints on how to move forward would be appreciated. I suspect one might come up with an plausible verbal argument to show the above. Hopefully I have a well defined problem above.

Thanks for any help.

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# I Resultant vector field as sum of many sources

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