I have been mulling over for some time how Dr. Bussard came to the conclusion that power gain scales as the fourth power of the magnetic field times the radius and that power out scales as the fourth power of the magnetic field times the radius cubed.
I think I have the answer thanks to our number magician Indrek. The very first graph is I to v. The I is the coil current. The v is the velocity of the electrons. The coil current is directly proportional to the magnetic field. The energy of the electrons goes up as
1/2 m v2
Where m is the mass and v is the velocity.
That tells us that the energy of the electrons that the magnetic field can confine goes up as the square of the magnetic field. Since the density of electrons that can be confined also goes up as the square of the magnetic field and the density of the electrons determines the density of the two reactants power goes up as the fourth power of the magnetic field as does power gain since the densities are multiplied to get the power output. In addition power gain goes up for the same reason.
The second part of the power equation - r cubed - is easy to figure. The bigger the reaction volume the more power out.
So the last question is why does the gain go up as r ? Easily answered. Electrons only have a chance of escaping the reaction area if they hit a magnetic wall. At a given electron energy (it would be fixed no matter the size of the reactor) the time it takes to go from magnetic wall to magnetic wall depends on the distance the walls are apart i.e. the size of the reactor. Bigger reactors inherently have lower losses or to put it another way - higher gains.