4 charges at vertices of regular tetrahedron?

Question

What will happen if 4 point charges +1,+1,-1,-1 initially at rest
at vertices of regular tetrahedron are simultaneously released?

Answer 1

Off the top of my head, I would expect the charges to cycle between tetrahedron extremes, passing through a stage where the charges are co-planar in a square formation, of sides smaller than the tetrahedron extremes. Such an "orbiting" system of trajectories would be highly untstable, and very likely to lead to chaotic trajectories where 2 pairs of unlike charges will combine, shooting off in opposite directions from the original center of the tetrahedron.

Answer 2

Revised for readability: enable a, b, c, d ? ?² be the respective projections onto the first 2 coordinates of the vertices ?, ?, ?, ? of a universal tetrahedron sitting in ?³. observe this projection is orthogonal. through translating by way of their mean, we may be able to anticipate ? + ? + ? + ? = (0,0,0). It follows that a+b+c+d = (0,0). Denote their negatives as -a = a', -b = b', -c = c', -d = d', that are projections of -?, -?, -?, -?, respectively. Now all 8 factors at the same time are the orthogonally projected vertices of a dice sitting in ?³. we may be able to exhibit those 8 factors as all ± combos ±u ± v ±w the position u = (a-b')/2 = (a+b)/2, v = (a-c')/2 = (a+c)/2, w = (a-d')/2 = (a+d)/2. observe that u, v, and w are projections of (? + ?)/2, (? + ?)/2, and (? + ?)/2, that are at the same time orthogonal and all a similar length (Use ? + ? + ? + ? = (0,0,0)). enable D be the three-through-3 matrix whose rows are those row vectors (in that order). observe D is an orthogonal matrix (interior the experience that DD? is a scalar dissimilar of I). Now enable C be the three-through-2 matrix whose rows are the row vectors u, v, and w, in that order. observe C is in simple terms the first 2 columns of D. It follows that the columns of C are orthogonal (as column vectors) and performance a similar sq. modulus. it isn't complicated to work out that this mandatory condition (that the columns of C are orthogonal as column vectors and performance a similar sq. modulus) is likewise a sufficient condition for there to exist a universal tetrahedron of whose vertices a, b, c, and d are the projections: certainly, the 0.33 column of D might want to be taken to be the normalized flow manufactured from the columns of C. The vectors a, b, c, and d might want to be recovered from the rows of C because the combos ±u ± v ± w having a good variety (0 or 2) of minus warning signs.

Answer 3

WHaaaaa??!