vdg voltage
How to measure the voltage on the VDG? You can't connect a meter, since anything touching the sphere causes discharges. The breakdown voltage in air depends on many things, including the shape of the grounded object. I had observed that my discharge sphere (B on this picture) moves every time there is a spark, which means there is a noticable force that attracts it to the VDG dome when it is charged up.

Based on that, I built the contraption shown on the right. The grounded sphere is suspended from one arm of a sensitive balance, and a counterweight is suspended from the other end. The counterweight is a can filled with enough screws to balance the sphere, plus a few (Order 10g) grams more. The can rests on my electronic kitchen scale.

In the picture, I have highlighted the (thin, non-stretch) string that runs up from the sphere, over two small nails on the balance arm, and down to the can, where it is tied with a simple knot.
The balance is an inverted V, where the fulcrum is a boxcutter blade (highlighted in green). The inverted V is self-leveling if there is nothing attached to it. This means that under load, there is a small restoring torque unless the arm is level. In this picture you can also see the thin wire that comes up rom the small sphere, and connects it to ground. coming up from the sphere

The measurement procedure is as follows:

  1. Untie the knot at the can, and re-tie it higher or lower. This raises or lowers the grounded sphere.
  2. Level the balance arm such that the horizontal part of the string is parallel to edge of the roof.
  3. Zero the scale
  4. Turn on the VDG, and see the scale readings change to a negative value
  5. When the reading stabalizes, record the value.
  6. Discharge the VDG, and measure the distance between the spheres.
Because the arm is always in the same position when a measurement is made, and the kitchen scale also doesn't move during a measurement (presumably it works with strain gauges, and not with springs), many systematic effects (such as the small force, if any, from the grounding wire) are canceled out.
Measurements: I untied and retied the counterweight a bunch of times at different heights, 
   distance between   scale
   the spheres (mm)  reading
   -------------------------
        77    spark at 20g
       116    spark at 16g
        84             12g
       147             10g
       184              9g
       184              7g
       159              9g
        95             17g
        62    spark     -
        85    spark     -
       119(?)          17g
   -------------------------
Ambient conditions on 28 Dec 2014: 32°F, 33% relative humidity, sunny, no wind.
Big sphere: 11" diameter → R1 = 14 cm.
Small sphere 38 cm circumference → R2 = 6.0 cm.
plot script
What voltage would cause this force? I found some papers that contain expressions for this:

1) Cruz and Ley-Koo: Evaluation and measurement of forces between two conducting spheres

This paper had calculations as well as measurements. I overlayed plots for different ratios between sphere radii. Our spheres have a ratio of 0.43, which turns out to be a very large extrapolation from the published curves.


The authors very kindly ran their code to produce a new curve corresponding to spheres of 1.775 and 0.76 cm, which is the same ratio as my spheres. In blue is plotted a subset of the new points.

and
Lekner: Electrostatics of two charged conducting spheres

and
Electrosttic interaction between two conducting spheres

and
Lekner: Electrostatic calibration of sphere-sphere forces

and
Lekner: Electrostatics of two conducting spheres

and Electrostatic force between two conducting spheres at constant potential difference

xxx
Hubert Van Hecke
Last modified: Mon Jan 19 19:49:02 MST 2015