TallDave wrote:Sure. Now turn one 180 degrees and place it next to the other. Is there a null at the center between them? Or do they make one big magnetic field that an electron bounces off of as though it were one big coil?
It gets... complicated and hard to visualize.
Magnetic field lines never end and they don't cross; they are (directed) loops. Around a straight conductor with a flowing current, they form circular rings. Bend that conductor into loop, and they bunch together in the center and spread out around the outside. Put more loops, or higher current (effectively the same thing), and you get more field lines through the loop, for a stronger field. (The density of the field lines is proportional to the field strength.) Any straight line which isn't straight through the bore will cross many field lines.
If you put two solenoids side by side, look at where the field lines can go. They have to remain loops. They can't cross. If two solenoids are side by side with N pointing in the same direction, the field lines coming out of the bores along the shared side can't curve far away from the solenoids to form a loop because of the other field lines going in the same direction, so they end up curving close and tight to the solenoids, forming a strong field going the other way between the solenoids. In this case, there is a plane, equidistant from both solenoids, in which no field lines cross and all close field lines are directed anti-parallel to the ones in the bore.
If two solenoids are side by side with N pointing in opposite directions, the field lines emerging from the bores along the shared side have an easy way to form a loop: just go back through the other solenoid! As such, you have a significant portion of the magnetic field lines passing through both solenoids in a closed loop, and virtually no magnetic field actually between the two solenoids. In this case, there is a plane, equidistant from both solenoids, in which all field lines which cross it are perpendicular to it.
As for electron confinement, the key is that electrons are deflected when they cross field lines, not when they travel parallel to them. In the situation above (two equal solenoids side by side with bores parallel to and equidistant from the x axis) an electron in the plane equidistant to the two solenoids would not cross any field lines when approaching two solenoids with N pointing in the same direction and would see no deflection. In the other case, an electron travelling in that plane would cross a metric buttload of field lines when approaching two solenoids with N pointing in opposite directions and would see a lot of deflection before it got close to the area between solenoids where there is little to no actual field.
We have all these places along the line edges of the cube where magnetic fields run into each other. Aren't they butting against each other and making cusps?
Yes, they are! As you have gathered, that's not good. If you have adjacent solenoids who's bores are both pointing in the same direction, you'll get planes between them with no deflection. Typically, we think of a surface containing the polywell, and these planes intersect the surface along lines, so we call them "line cusps", and they are a loss mechanism. They are an artifact of the shape and design of the coils. If the coils on the WB-6/7 were square instead of round (and mounted corner-to-corner, like the square faces of a cuboctahedron) then the line cusps would be greatly reduced or eliminated.
Another set of cusps are along the bores of the solenoids, where you have strong field lines going directly parallel to the radius from the center and thus no deflection. We call these "point cusps", and they are less of a concern than the line cusps because they are generally smaller (a point instead of a line).
There's also something called a "funny cusp", but I'm not 100% sure what that is.