A brief note on safety
A Van de Graaff Generator is capable of generating extremely high voltages, but it is very safe due to the very low current, and quick dissipation of the arcs. For almost all people, the vdg will not adversely affect you in any way (I can't say the same for your sensitive electronics), but if you have a bad heart, it would be a good idea to steer clear of a Van de Graaff generator.
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Step 1: How it works
Well it just so happens that the Van de Graaff generator works on a very simple principle. I'm sure you are all familiar with what happens when you walk on carpet with rubber shoes on a very dry day. Your hair stands up, and you are able to send electricity arcing through the air to the dismay of anyone arms reach away from you. This occurs because of something known as triboelectric transfer. When you walk on the carpet the carpet will exchange a charge with you. As you continue walking the charge continues to build until your body does not have the capacity to hold anymore charge, or you get too close to something that has a difference in potential and all the charge suddenly leaks from your body through a path of ionized air.
A Van de graaff generator takes advantage of triboelectric transfer by using a belt on two rollers to carry a charge from a metal comb to another metal comb where the charge gets transferred to a metal sphere.
In construction, the vdg is extremely simple, but the principle that it works on is far more complex. For example, these generators do not have their belt rub on the rollers as rubber shoes do on floors. The triboelectric transfer occurs simply through contact not friction. This does mean however, that the type materials that the vdg belt comes in contact with on the rollers is crucial to opperation. To be effective the materials of the two rollers should be on opposite ends of the triboelectric series, and the belt should be made from a material that falls somewhere in between them in the series. Triboelectric series is nothing more than a list of materials ordered by their triboelectric effect, or the way they exchange charges with other materials. Since the vdg relies on this affect if the two rollers are triboelectrically identical no charge can be built up. In fact, the orientation of the rollers is what determines the polarity of the output. In my case I used teflon for the bottom roller and aluminium for the top so I got a positive output. Swapping the rollers would give me a negative output.
The rollers and belts are only part of the story though. If you look at the diagram above, you can see that two metal combs or brushes are used on the machine. the bottom brush is grounded, and removes the charge from the the top electrode. The top brush transfers the charge from the belt to the top sphere. To better understand what happens with the brushes it would be better to think about an example. Imagine a vdg with a positive output. as the belt comes down from the top roller to the bottom roller, it is carrying electrons from the top brush. As it approaches the bottom brush, the charge is sufficient to ionize the air between the brush and the belt and the electrons move to ground. as the belt continues along, it is triboelectrically charged by the bottom roller to be positive. As it goes across the top roller it begins to pull electrons from the top sphere, leaving unpaired protons, and therefore positively charging the top sphere.
The top sphere of the machine itself is actually highly important to the output of the vdg for many reasons. For example, with the above parameters there is no reason why a vdg couldn't continue building up a charge forever, that is if it wasn't for the fact that for one thing there are only so many unpaired protons that could be left on the sphere, and because any kind of small imperfection or point on the sphere would begin to cause corona discharge, where the charge literally leaks off the sphere and into the air causing a large amount of losses. That being said, The bigger, rounder and smoother the sphere on the top of the vdg, the higher the output it is capable of reaching.
I certainly haven't covered everything, and there is a lot more going on in the operation of vdg's, but hopefully now you understand the basics of how they work.