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A lot of practical high voltage design requires knowing what the maximum E-field is, for insulation design, corona reduction, etc. The exact field can, of course, be calculated numerically by solving Laplace's equation over a suitable field with appropriate boundary conditions. As complicated and time consuming as this is, it is necessary when performance is critical, in integrated circuit design, designs for absolute minimum cost, and so forth. However, for more run of the mill experimentation and use, where a little overdesign can be tolerated, approximations to the field are just as useful.
The mean or average field is just the voltage difference divided by the distance between the conductors. For the proverbial infinite flat plates, this makes the calculation simple.
Emax = Eaverage = V / Distance
For two concentric cylinders (i.e. like coaxial cable) the maximum field is:
Emax = V / ( Rinner * LN( Router / Rinner))
For concentric spheres, using the same variables, the maximum field is:
Emax = V * Router / (Rinner *(Router - Rinner))
For two parallel cylinders of equal radius:
Emax = V * SQRT(D^2 - 4 * R^2)/ (2 * R*(D-R)*INVCOSH(D/(2*R)))
approx equal to: V / (2*R) * LN(D/R) if D>>R
For two spheres:
Emax = approximately V/S * F
F = (S/R+1) * sqrt( (S/R+1)^2+8)/4
For spheres, if S>>R then Emax = approx V/ (2*R)
For other configurations:
Emax = Eaverage * F
For sphere/plane: F = 0.94*Spacing/Radius + 0.8
For cylinder/plane: F = 0.25 * Spacing / Radius + 1.0
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