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Re: [Phys-l] [phys-l] electric field due to uniform ring of charge, off-axis



On 1/12/2011 3:52 PM, Krishna Chowdary wrote:
Dear colleagues

Is there a closed form expression for the electric field due to a
uniform ring of charge (charge Q, radius a) at points in the plane of
the ring (but not on the ring itself), or even more generally anywhere
in space (but not on the ring itself)? If yes, I'd appreciate
suggestions on how to proceed.

It's a standard exercise to determine the field for such a ring on its
axis. What about not on the axis?

I can do this numerically with a straightforward spreadsheet
implementation. I can't yet do it analytically (I couldn't find an
elementary anti-derivative, and Mathematica and WolframAlpha invoke
the Appell Hypergeometric Function of two variables, which is far
outside my experience).

If you're interested, here are the integrals I'm trying to evaluate,
where (x,y) is the coordinate of the field point and a is the radius
of the ring (centered on the origin):

http://tinyurl.com/4c5fp3x

http://tinyurl.com/4gnmd99

I appreciate your insights.

sincerely,
Krishna
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l

Seems to me it may be better to account for the other end (so to speak)
of the electric field. I am thinking in particular of the electron meniscus
doublet lens formed by a disk, a cylinder, and another disk along the
flight axis of the electron beam.
In that arrangement (say an electrostatically focused oscilloscope etc).
forming an electron lens, the disks are connected to one potential
and the cylinder is connected to a lower potential.

Brian W