Electric field on a tapered cylinder?

[HMS-776]

Electric field on a tapered cylinder?

I have a rod which is 1" in diameter, and 4 inches long, the last inch of one side of the rod tapers down to a point of 1/10th inch (.01).

The rod also has a cylinder which it fits in, which has a taper which follows the rods taper exactly. There is a gap between the rod and cylinder through the whole length of .01

If a voltage were applied to the large diameter end of the inner rod of +100V. And the (outer) cylinder was grounded, What would the voltage be at the point which is 1/10th the diameter of the large end?

Since the rod and cylinder essentially form a capacitor, would the smallest area (tapered point) have the largest charge? Would the charge be magnified, or greater than the applied charge because of the reduced area?

How could the charge at the point be calculated?

 

[Born2bwire]

My answer to any general calculating charge question is to use a moment method. There are usually closed form approximations you can make but since you are interested in the behaviour of the field around the taper then I do not know of any easy way of doing it via closed form. Maybe there is a way, I am not aware of it. But the point is going to give rise to field enhancement. The charge density there is going to be higher due to the smaller surface area and this is going to increase the electric field local to the point.

 

[Vanadium 50]

There are closed form - or nearly so - solutions, but they are far from trivial. See Section 3.4 of Jackson.

I sure hope you like hypergeometric functions!