Robthebob wrote:
B field vector components sum up appropriately. So because field strength of ring current changes as you move away from the ring, it's not constant, and it's not zero anywhere inside the ring and the ring plane, it's impossible for opposing magnets fields to cancel each other at the center of the ring in the ring plane. In fact the only place the field would be low or zero is at the very center of the polywell, where all opposing rings' fields cancel each other out.
You are essentially saying that opposing magnets in a 3 D arrangement can cancel out, and that this cannot occur in a 2 dimensional arrangement (space). And yet you (and/or others) use a two dimensional representation (2 D slice) of the Polywell, point to it and say- 'See the zero field in the center'. With the above argument, this is nonsense.
There seems to be confusion about digital and analog representations of magnetic fields. In a vacuum magnetic fields are analog in nature, at least in the ranges we are talking about. At some intense scale quantum effects become significant, but we are far from those conditions in the Polywell so it is safe to say the B fields are approximately analog. They decrease to zero at an infinite distance from the source. With opposing magnets, whether arranged in a 2 dimensional plane or a three dimensional sphere, the same applies except that the opposing fields, well... oppose each other, They do not pass through each other (which would imply that they were not opposing). There has to be a zero field strength between the two domains. Granted this is a theoretic situation in a pure vacuum and practically this condition could not occur in the real universe (which is not a pure vacuum). But, it is safe to say that so long as the conditions where quantum magnetic effects and uncertainty principles are insignificant, the zero condition is very closely approximated. Also, granted this almost zero B field strength region is much smaller than the gyro radius of any charged particle of interest, still it exists. As I have repeatedly stated the important point is the gradient at which the B field increases from this null point (or near null point if you prefer) to strengths that effects the particle containment - mirroring. This is a moving target depending not only on the B fields, but also the charged particle characteristics- speed and vector. The cusp diameter, area is thus defined. You could define a cusp as extending to near the surface of the magnet cans, it all depends on the limits you choose. I don't know what standards are used. An assumption may be that a cusp diameter is chosen that results from 99% of all possible charged particle trajectories/ vectors results in mirroring. Any other assumption may be used.
The difference between a true point cusp and a line cusp, is that a point cusp loss area will be a circle or other closed geometric shape , ideally the length is ~ the same as the width. A line cusp has a length that can be unlimited theoretically. Piratically, in a circle or other closed geometric shape the length is limited by the circumference. Also, a line cusp by definition has physical surfaces bridging it if magnets are connected to each other by nubs, or long nubs/ standoffs. These can be partially shielded but cannot be shielded as well as the main magnets. ExB and possibly significant ExY drift issues are relavent. At equal intensity relationships a point cusp loss area may be 1 cm wide by 1 cm, long while a line cusp would be 1 cm wide by perhaps 31.4 cm length with a 10 cm diameter magnet ring (actually a pair of such rings where the separation is equal to the diameter of the point cusp generating magnet ring).
What is significant about the Polywell is that the separation of the line cusp generating rings are moved much closer together, thus the fields are stronger in the mid section (correction- near the mid section, I almost ignored my own perspective) between them and this significantly reduces the width of the cusp loss area. This can be simply considered as a stronger central field, but is, again, the gradient from the null center to the adequate field strength that determines the loss area, cusp width. As the magnets are moved together this width decreases till eventually the product of the width times the length of the line cusp can approach or even improve on the loss area of the corresponding point cusp. This is what is meant by the line cusps becoming "point like cusps" . The physics of the line cusp does not change , only the "practical cusp loss area". A line cusp does not become a point cusp.
The obvious consequence of moving the magnets closer together is two fold. The line cusp losses become less, but the internal volume also becomes less, and fusion yield scales accordingly downward. The Polywell is a cleaver 3 D arrangement of magnets that greatly reduces the width dependent loss component of the unavoidable line cusps without the penalty of decreasing the effective volume. This is why the total cusp loses in the Polywell at the same internal volume of an opposing two magnet mirror machine is about 10 times better (taken from the EMC2 patent application). This is significant, but still not enough. The further improvement from the Wiffle Ball effect makes up the rest of the gains, but that is a different topic.
A line cusp is a line cusp. I do not see how placing a point cusp (even if the x-cusp can be considered a point cusp)between two line cusps would improve things. he key is to move the line cusp generating magnets as close together as possible. Bussard did this by having the magnets touch, but he had a flaw in his reasoning that negated ExB drift concerns. He reconized that he still had line cusps, but he ignored the unshielded losses by assuming the magnet cans were theoretical lines so that the aviable loss area exposure was infinatly close to zero. Thus the "Funny cusps where recognized line cusps but acted effectively like point cusps (ExB drift and ExY drift concerns were inconsequential). Of course WB6 with it's spacing between magnets was a recognition of the real world versus the theoretical one. There was a needed compromise between cusp mirroring loses and ExB drift loss concerns. The ~ 10 fold improvement in WB6 electron containment reflects that the ExB losses (along with conformal can shape and nub minimizing) were significant to such an extent that adjusting the cusp mirroring confinement efficiency in order to reduce the other losses still resulted in a net gain (with the help of increased recirculation). With X cusps you do shorten the line cusps some, but you are not eliminating the exposed metal. And the line cusps in the basic truncated cube Polywell already has line cusp confinement properties better than the true point cusps. Ignoring ExB drift concerns, I concede that relevant B field strengths (gradients from center to magnet surfaces) would help within the X cusp, but I'm doubtful there would be a net gain.
If available, the comparison between WB 7.0 and 7.1 would be enlightening.
Back to the analog versus digital perspective. KitemanSA likes to point out that the isobars represents a null field in the center of the near sphere but still ignores these isobars extending into and through the cusp. You can choose a resolution that crowds the iosobars so close together in the cusps that they look solid, but this is a digital artifact. It does not imply that the fields are stronger in this region (compared to the same isobar in the center of the machine), it means that the gradient is greater. This is the same as weather pressure maps or land elevation maps. The isobars do not imply strength or height directly. They have to be labeled or traced to a section of the same isobar that is labeled to determine strength (either that or count the isobars against a defined strength scale which is the same thing). What they do imply is that as they become closer together the gradient or rate of change becomes greater. In the Polywell representations this is the weakening of the magnetic fields as they approach the midline or mid point between opposing magnets. It the illustration that I'm sure KitemanSA used the field lines are very close together in the midplane of the magnets, compared to the center of the machine. This implies that from the center of the cusp towards one of the magnets, the field strength increases rapidly to the point where the cusp boundries are defined. This implies a smaller cusp area, but does not have anything to do with the field strength in the dead center of the cusp (which is zero or very close to it). This could be interpreted as having stronger fields in the center, but this is misleading, though perhaps practically useful. It is all about the gradient to some limit that is reached at some isobar away from the center/ mid line. This is exactly the same as the central region of the three dimensional near sphere. At some strength (isobar) the charged particles will be mostly mirrored. Because the isobars are further separated in the center (because it is further away from the magnets) the surface area/ volume of this area is greater than the cusp loss area at the magnet mid planes. There is still B fields in the center (except at the exact center), but the strength is too weak to mirror. Of course the picture changes when significant numbers of charged particles are introduced and Wiffleball effects become significant. But for this discussion the magnetic field strength in an absolute since is not important. It is how far out the isobar that is linked to mirroring is pushed out . I suspect the same occurs in the cusps, but surface area change ratios are much different. ie- the central area increases considerably, but the mid plane cusp loss cross section increases much less because the neighboring B field gradient is greater.
Dan Tibbets
To error is human... and I'm very human.
