. V AN D E G RAAFF G ENERATOR This is a machine that can build up high voltages of the order of a few million volts. The resulting large electric fields are used to accelerate charged particles (electrons, protons, ions) to high energies needed for experiments to probe the small scale structure of matter. The principle underlying the machine is as follows.
Suppose we have a large spherical conducting shell of radius R , on which we place a charge Q . This charge spreads itself uniformly all over the sphere. As we have seen in Section . , the field outside the sphere is just that of a point charge Q at the centre; while the field inside the sphere vanishes.
So the potential outside is that of a point charge; and inside 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 Q ( . ) Now, as shown in Fig. .
, let us suppose that in some way we introduce a small sphere of radius r , carrying some charge q , into the large one, and place it at the centre. The potential due to this new charge clearly has the following values at the radii indicated: Potential due to small sphere of radius r carrying charge q at surface of small sphere at large shell of radius R . ( . ) Taking both charges q and Q into account we have for the total potential V and the potential difference the values ( ) Q V R ( ) Q V r ( ) – ( – V r V R ( .
) Assume now that q is positive. We see that, independent of the amount of charge Q that may have accumulated on the larger sphere and even if it is positive, the inner sphere is always at a higher potential: the difference V ( r )– V ( R ) is positive.