# Point-Plane Electric Field Intensity

1. May 1, 2007

### rpiengineer

Hi folks. I have read in several papers that the electric field intensity adjacent to the very tip of an electrode (the "point" in a point-plane geometry) surrounded by a dielectric medium is given by

2V / [r ln(4d/r)]

where r is the tip radius, d is the point-plane spacing, and V is the applied voltage. The papers I have read do not give a reference for this equation, however. Does anyone know where this equation comes from? I have thought about different ways one might derive this, but my models are lacking in their use of the electrode radius "r". The undergraduate and graduate textbooks I have available do not address this particular problem.

Many thanks in advance to anyone who can help me here !

Rob

2. May 3, 2007

### waht

That's kind of weird, why would you need a radius of a fine point tip? If it's a tip then you could assume it's just a point with 0 radius, but obviously that's not the case, so unless there is a trick but I don't see one right now, other than to use nasty double or triple integrals in polar coordinates. Or solve the laplace equation, but the boundary conditions would kill you.

I will try and look into it more.

3. May 3, 2007

### rpiengineer

Thanks so much. I know, having done endurance testing on polyethylene with embedded needles, that the tip radius is the most important factor in knowing the highest field intensity. I suppose this is because the smaller the radius, the more divergent the field, thus the higher the field intensity in the dielectric right next to the metal tip. We measure the tip radius of etched electrode needles before we insert them into molten polyethylene, then cool it to room temperature for testing. I just don't know how the many authors who make use of this relationship arrived at it. I would like to include the rationale - if not the entire derivation, which may be a bit long - in my dissertation.

Rob

4. May 3, 2007

### mdelisio

Not sure if this helps much, but the form of your solution is very similar to the fields around a wire of radius r parallel to a ground plane a distance d away. If the potential separating the wire and the plane is V, you get that the E-field at the wire is approximately:

E = V/(r*ln(2d/r)).

When I do it with a conducting sphere above the plane, I get

E = 2d*V/(r*(2d-r)),

however, which does not have the same form...

Last edited: May 3, 2007
5. May 4, 2007

### rpiengineer

I agree, the logarithm must come from the circular tip shape somehow. Hmmmm...not sure what to think at this point.

Thanks for looking into it for me !