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Curtis Broadbent, PhD.
Oct. 9, 2018
So you have a Van de Graaff generator and you want to know how good it is? (Yes, that’s two ‘a’s and two ‘f’s after Robert J. Van de Graaff, the inventor). Well, turns out the proof is in the electrical pudding, or rather the electric arcs the Van de Graaff generator can create. I’m not lying! The voltage of a Van de Graaff generator can be measured by its ability to shock you!
Here’s the long and the (electrical) short of it. The air around us usually acts as a pretty good insulator. 9V batteries, though they give a good zap on your tongue, don’t self-discharge in air by current flowing between the electrodes. This is because the air-gap prevents the charge from flowing between the positive and negative electrode. But if you increase the voltage from 9V to something much much higher, the electric field between the electrodes stretches the positive and negative parts of the atoms and molecules in the gap until – POP! – electrons are ripped off and go careening toward the positive electrode while the ions go in the other direction! These electrons and ions don’t just fly through free space, though, they fly past other atoms and molecules sometimes colliding into them. Those collisions help ionize other atoms and molecules, and the resulting electrons and ions collide with yet other atoms and molecules, perpetuating the ionization and current flowing process. This quickly snowballs, resulting in an electrical arc between the positive and negative electrodes.
One might think that the relevant key question is “How large does the potential difference have to be before arcing occurs?” Actually, this is not the correct question; it is the electric field, not the voltage between the electrodes that is responsible for creating the arc. And the electric field depends on the voltage and the geometry of the electrodes. The electric field will be different if the electrodes have pointy tips or if they are flat like the sides of a parallel plate capacitor. Also, whether they are far apart or close together will influence the magnitude of the electric field. Remember the parallel plate capacitor? In this case, the electric field is uniform and has a value of E=V/d. This tells us that increasing the distance between the plates results in smaller electric fields. Roughly speaking, this rule of thumb applies for all electrodes, even spherical ones like are used in the Van de Graaff generator and discharging wand.
So, here’s the procedure. Fire up the Van de Graaff generator and put the wand close to the large sphere so that arcing occurs. Then slowly pull the wand farther and farther away from the large sphere until arcing just barely stops. This distance between the wand and large sphere (we’ll call it the maximum arcing distance, ) is related to the minimum electric field which is able to create an arc. And that is directly related to the voltage difference between the wand and the large sphere. So if we know the maximum arcing distance we can determine the voltage, provided we have a formula relating these three quantities.
Unfortunately, simple formulas are hard to come by, except in the case of parallel plate capacitors at voltages less than about 100 kV. And, generally speaking, the electric field strength at which arcing occurs also depends on the humidity and air pressure, not just electrode geometry. Scientists and engineers have worked out physical models and have generated accurate look-up tables which have been condensed into an international standard that is accurate to 3% if followed precisely . However, for large spherical electrodes near room temperature and within the range of normal atmospheric pressures and humidity, the field strength at which arcing occurs is roughly equal to 3 MV/m. And while the electric field between the spheres is not uniform, it is close enough to uniform (see Figure 1 below) that the parallel plate formula quoted above can be used to derive the following approximation for the Van de Graaff generator voltage,
where d is measured in meters. This formula is fairly accurate, and given its pedagogical simplicity, it’s a good compromise for educational settings . Accuracy can be improved by replacing the discharge wand with another Van de Graaff generator that has been grounded.
Figure 1. The field strength is highest and most uniform between the two spheres. Yellow represents positive charge and blue represents negative charge. Can you guess which sphere is grounded? Plot was generated using a fantastic java applet by Paul Falstad available online at www.falstad.com/emstatic.
You may have seen an approach for measuring the voltage of a Van de Graaff generator based on the electric field at the surface of a charged sphere and the capacitance of a spherical conductor [3, and references therein]. This approach is less accurate than the arcing measurement because it assumes a perfect sphere with no imperfections. In practice, the voltage attainable by the Van de Graaff generator is limited by the leakage current from the dome which is mostly a function of imperfections to the sphere. These imperfections can take the form of the 4 mm banana connector receptacle often placed on top of a Van de Graaff generator dome, the truncated spherical shape common in Van de Graaff generators, and imperfections in the smoothness of the sphere surface. The arcing-based measurement, since it is based on the ability of the actual voltage to generate an arc, indirectly takes all these imperfections into account.
Finally, a hint. The best tool for measuring the distance between the spheres is called an internal, or inside caliper (see this caliper, for example). If you’re like me and don’t have one of these in your tool kit, you can simply take two popsicle sticks and pinch them together in a ‘V’ shape in the gap to get the distance. Then hold the sticks against a ruler to get an accurate gap measurement. Be sure you don’t let them slip when you transfer to the ruler. And to avoid an unpleasant shock, be sure you turn off and discharge the Van de Graaff generator before you take the measurement!
 International Electrotechnical Commission. (2002). Voltage measurement by means of standard air gap (Standard No. IEC 60052:2002). Available from https://webstore.iec.ch/publication/290, accessed Oct. 9, 2018.
 SSERC. (2007). Van de Graaff generator hazards (Bulletin 223). Available from http://info.sserc.org.uk/bulletins226/2007/223-winter-2007/1220-van-de-graaff-generator-hazards235, accessed Oct. 9, 2018.
 SSERC. (2002). Van de Graaff generator hazards (Bulletin 205). Available from http://info.sserc.org.uk/images/Bulletins/205/4-6_Van_de_Graaff_generator_hazards.pdf, accessed Oct. 9, 2018.