Thanks, Horace, this was enlightening. Is it fair to say that,
a) The Casimir force is a surface phenomenon, unlike most common forces, which act on the body of the material, including all the forces mentioned below in (b); b) The net Casimir force which acts on a body due to the presence of more than one "shading" object is *not* related to the force due to each individual "shading" object by linear superposition, unlike EM forces, Newtonian gravity, and push gravity (in the low-strength limit); and for these reasons, and most particularly due to (b), the Casimir force is also *not* conservative, unlike the other common forces mentioned in (b)? Linear superposition is what allows us to conclude from the conservative nature of single-particle E and B fields that the action of arbitrarily complex E and B fields must also be conservative, of course, and the fact that the Casimir force doesn't follow this is remarkable. Horace Heffner wrote: > [ snip parts I didn't comment on ] > > I am well aware of the fringe force at the edge of capacitors and how > these are used to produce electrostatic motors. Electrostatic motors > conserve energy, so it just happens that when an edge with a small > fringe force is dragged across a wide capacitor, a lot of energy is > expended (or obtained depending on polarity) and that energy is exactly > equal to the energy stored between the plates when together at > potential. This involves the Coulomb force, which is a 1/r^2 force. I > have also been aware that the Casimir force involves large negative > exponents on r, and thus the fringe force effect can not be conservative > with regard to plate effects. Don't follow this particular paragraph's conclusion, which *seems* to be saying that because it's higher power than 1/r^2, it can't be conservative. But perhaps I misunderstood this. Just because it's higher power than r^2 doesn't imply it's not conservative, does it? Surely, for instance, a 1/r^12 force can be represented by a potential field going as k/r^11, can it not? But in any case this has no bearing on what you said both earlier and later, which made it seem clear the Casimir force should, indeed, *not* be conservative. > Further, and most importantly, the casimir > force involves only straight lines. There are no bending E field lines > to deal with. Right -- I think this is a direct consequence of the fact that it doesn't obey the law of superposition. The E field consists of nothing but straight lines, also, at some fundamental level, but when we superpose several sources which all produce "straight lines" with magnitudes varying along the lines, the *sum* of those lines is a bunch of curves. > The energy involved in plate motion with regard to the fringe force, at > least for conductive surfaces, is highly dependent on the plate edge > geometry. It is purely an effect between surfaces. While > Mostepanenenko's principles involve fudging, he says the powers of r > always come out right using his method. Therefore the method serves > well for a comparative analysis, because, in designs using the same > materials, we can ignore the material related (fudged) constant C. I > will now prove, by comparative analysis, that the lateral force on a > square plate edge is not the same as on a beveled plate edge, and thus a > net energy gain is feasible from a Casimir effect motor provided the > edges of the plates are appropriately shaped. > > The following comments refer to aspects of Figure 6, Page 9 of: > > http://www.mtaonline.net/~hheffner/CasimirGenerator.pdf > > The following is the key to Fig. 6: > > (P7,S3) - top of bottom slab > (P5,P1) - bottom of top slabs > (P5,P1,P2,P4) - sides of rectangular top plate > (P5,P1,P3,P4) - sides of beveled top plate > S1 S2 and S3 - points for van der Waals retardation calculation > (S1,S2)- ZPF free line for S1 on beveled slab to some point S2 > S4,S1,S5 - angle of feasible ZPF free lines to second plate from S1 > (s3,S2) - SPF free line corresponding to (S1,S2) for comparison > > In Fig. 6 we are comparing two slabs, a rectangular one vs one with a > beveled edge. Now we pick any point S1 on the edge of the beveled slab. > We need to sum the potentials (or forces) from every ZPF free line > between S1 and every point S2 on the bottom slab surface. All such ZPF > free lines (S1,S2) lie within the angle S4,S1,S5. The line (S1,S4) is > parallel to the bottom slab surface, (P6,S2). > > Now, for every such ZPE free line (S1,S2) there exists a unique > corresponding ZPF free line on the rectangular slab edge (P1,P2), which > shares an intersecting point S3 on the rectangular slab edge. In every > case, (S1,S2) is larger than the corresponding (S3,S2). Because this is > true for every possible point S1, the sum of the potentials for the > beveled edge are smaller than the sum of the potentials for the > rectangular slab edge. There are many ZPF lines from the rectangular > slab edge to the bottom slab which have not been evaluated, and for > which there are no corresponding ZPF free lines from the beveled edge, > but this only makes the inequality larger. In every case of > corresponding ZPF free lines, the lines make the same angles with > respect to the bottom slab, so the force vectors can be broken into axis > oriented vectors and summed without changing the direction of the > inequalities. > > There are analogous arguments if the bevel goes the other way, making an > even better cavity than the rectangular slab edge. > > For the above reasons, a free energy rotor can be made of blades of > parallelogram shape, with a back bevel on one side and a forward bevel > on the opposed edge, or even a squared off edge substituting for one of > the beveled edges. Since the force on the two edges is not balanced, a > net torque results without application of external energy. This results > in a free energy motor. Very nice! Nice demonstration that the force isn't conservative, and I like the rotor, which leaves the plates at fixed separation and just has one of them spin -- it's a lot more convincing (to me) than general assertions that we can bring the plates together along one path and separate them by sliding along a different, longer, path. > > This also provides proof that the earlier proposed Casimir generator > principles work, provided the slab edges are beveled. > > Best regards, > > Horace Heffner > http://www.mtaonline.net/~hheffner/ > > > >
