Electric field due to a charged ring off-axis

[a789irfan]

How can i find the electric field due to a charged ring off-axis? Actually i have two ring electrodes in parallel and i would like to know how to find the electric field at any point in between the ring electrodes?

 

[Meir Achuz]

This is treated in advanced EM texts.
The steps for one ring are:
1. Find phi on ths axis.
2. Expand phi in a Taylor series for either z>a or z<a.
3. Use this expansion to write the Legendre polynomial expansion for all
angle. Use phi=\sum r^L P_L(cos\theta) (or 1/r^{L+1}).
4. Take grad phi to get E.
The result is a bit messy for two rings, but it can be done.