| MadSci Network: Physics |
Hi Mark,
First of all: be careful when working with high voltages. Homemade Van de Graaffs are not particularly life-threatening; however, their worst shocks are extremely painful to anyone, and could kill someone wearing a pacemaker. The links below discuss some safety issues.
A Van de Graaff generator is basically a big capacitor. You put a lot of excess charge on a big sphere, and the sphere will be at a high voltage with respect to the rest of the world. This is just like a familiar two-plate capacitor, in which you put charge on one plate so that is stands at a high voltage with respect to another plate. There are really only two questions:
The first question is answered by a simple calculation: if you put a charge Q (in coulombs) on the sphere, the electric field anywhere outside of the sphere (a distance r from the center) is:
1) E = Q/(4 pi epsilon r^2)
and the voltage at the surface is the (negative of the) integral of E from r=infinity to r=R, with R being the radius of the sphere.
2) V = Q/(4 pi epsilon R)
The "capacitance" C is defined as Q/V, and tells you how much charge is needed for a given voltage. If the goal of your Van de Graff is to build up large amounts of charge, then you want a large radius, to get the largest possible capacitance with the least voltage. On the other hand, if you want large voltages, you'd want a smaller capacitance, so you don't have to wait for long while you pile up charge.
The equations above also contain the answer to "What is the limiting voltage of a Van de Graff?" To begin with, why is there a limit at all? The problem is with the electric fields. Whenever an electron (or any charged object) sees an electric field, it will want to move along it until it goes away. Thus, when you charge up a Van de Graff, the excess electrons on the sphere will see a large electric field (equation 1) and want to jump off of the sphere and follow it. Fortunately, the air is a pretty good insulator, and electrons cannot move through it easily. So electrons stay on the sphere and maintain the charge there.
However, like all insulators, air cannot handle arbitrarily large fields. An electric field of 3 million volts per meter (3 MV/m) will cause the air to ionize and become a conductor. Once the air itself can conduct currents, all of the excess charge on the sphere will discharge through the conducting path. This is what initiates a lightning bolt or a spark. So: use Equation 1 to find out how strong the electric fields are near the surface of your Van de Graff sphere. You absolutely cannot generate fields higher than this without sparking. Notice that larger spheres can handle higher voltages, since their surface fields are smaller at the same voltage. "Professional" Van de Graff generators, like this one are very large, and are surrounded by a better insulator (like sulfur hexafluoride) in a large high-pressure tank,
In practice, sparks will begin considerably earlier - any tiny irregularity (scratches, dents, burrs) will create a stronger local field, and initiate sparks at lower voltages. Polish your sphere well! And make sure that your mechanical supports (anything solid touching the sphere) can handle high voltages as well. Most plastics, rubber, glass, ceramic should be fine.
So a Van de Graff of any size should be able to create sparks, make your hair stand on end, etc. Larger models will permit larger voltages and larger sparks. I'd guess that a 4 cm sphere will spit out sparks several inches long! It should also charge up faster than a larger model, and spark at lower voltages.
Hope this helps!
-Ben
Some links: http://www.amasci.com/emotor/vdg.html http://hypertextbook.com/eworld/vdg.shtml
PS. The constant "epsilon" in the equations above is 8.8*10^-12 farads per meter. (i.e. use units of coulombs for charge, meters for distance, volts for potential, and volts/meter for fields)
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