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www.GOALias.blogspot.comElectrostatic Potentialand Capacitance2.16 VAN DE GRAAFF GENERATORThis is a machine that can build up high voltages of the order of a fewmillion volts. The resulting large electric fields are used to acceleratecharged particles (electrons, protons, ions) to high energies needed forexperiments to probe the small scale structure of matter. The principleunderlying the machine is as follows.Suppose we have a large spherical conducting shell of radius R, onwhich we place a charge Q. This charge spreads itself uniformly all overthe sphere. As we have seen in Section 1.14, the field outside the sphereis just that of a point charge Q at the centre; while the field inside thesphere vanishes. So the potential outside is that of a point charge; andinside it is constant, namely the value at the radius R. We thus have:Potential inside conducting spherical shell of radius R carrying charge Q= constant= 1 Q4 π ε0R(2.78)Now, as shown in Fig. 2.32, let us suppose that in some way weintroduce a small sphere of radius r, carrying some charge q, into thelarge one, and place it at the centre. The potential due to this new chargeclearly has the following values at the radii indicated:Potential due to small sphere of radius r carrying charge q= 1 q4 π ε rat surface of small sphere0= 1 q4 π ε0Rat large shell of radius R. (2.79)Taking both charges q and Q into account we have for the totalpotential V and the potential difference the valuesVan de Graaff generator, principle and demonstration:http://amasce.com/emotor/vdg.htmlhttp://www.coe.ufrj.br/~acmg/myvdg.html1 ⎛Qq⎞VR ( ) = ⎜ + ⎟4πε⎝RR⎠1 ⎛Qq⎞Vr ( ) = ⎜ + ⎟4πε⎝Rr ⎠00q ⎛1 1⎞Vr ( )– VR ( ) = ⎜ – ⎟4πε⎝rR⎠0(2.80)Assume now that q is positive. We see that,independent of the amount of charge Q that may haveaccumulated on the larger sphere and even if it ispositive, the inner sphere is always at a higherpotential: the difference V(r )–V(R) is positive. Thepotential due to Q is constant upto radius R and socancels out in the difference!This means that if we now connect the smaller andlarger sphere by a wire, the charge q on the formerFIGURE 2.32 Illustrating the principleof the electrostatic generator.83