The electric field due to a charged ring off-axis can be determined using advanced electromagnetic theory. The process involves calculating the electric potential, φ, on the axis of the ring, expanding φ in a Taylor series for specific conditions (z > a or z < a), and applying Legendre polynomial expansion. The electric field, E, is then derived by taking the gradient of φ. For two parallel ring electrodes, the calculations become complex but are feasible with the outlined method.
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