# Equal point charges on the sphere

I have coded a simulation where $$n$$ equal charges are randomly placed on the surface of the conducting sphere. I added a damping term proportional to velocity so that the charges arrange in a stable configuration. Then the graphics of polyhedron with charges as vertices is displayed.

It works for every number of charges less than 78. It takes my computer around 20 seconds for 77 but for 78 or greater it just freezes.

Below is the code. Under each defined function there is a description of what it does

Is there a way to fix it? Thanks in advance.

NaključnaTNaKrogli[r_: 1] := {
{x, y, z} = {RandomReal[{-1, 1}], RandomReal[{-1, 1}],
RandomReal[{-1, 1}]};
If[x^2 + y^2 + z^2 < 1,
r {x, y, z}/Sqrt[x^2 + y^2 + z^2],
NaključnaTNaKrogli[r]]
}[[1]];
(*returns random point on the sphere of radius r, centered at {0,0,0}*)

l[vektor_] := Sqrt[vektor.vektor];
(*returns magnitude(length) of a vector*)
L[sez_] := l /@ sez;
Strešca[vektor_] := vektor/l[vektor];
STREŠCA[sez_] := sez/L[sez];
sez\[CapitalDelta]r[sezr_, n_, i_] :=
ConstantArray[sezr[[i]], n - 1] - Delete[sezr, i];
(*returns list of vectors from i'th charge to all other charges in \
sezr. n=Length[sezr]*)
F[sezr_, n_, i_, c_] := {
s = sez\[CapitalDelta]r[sezr, n, i];
c Total[s/L[s]^3]
}[[1]];
(*returns list of force vectors of other charges on i'th charge in \
sezr. c is  q^2/(4Subscript[\[Pi]\[Epsilon], 0])*)
Seza[sezr_, n_, c_, m_, k_, sezV_] := {
rstre = STREŠCA[sezr];
sezF =
Reap[
Do[
Sow[
F[sezr, n, j, c]
]
, {j, n}]
][[2]][[1]];

(sezF - rstre (Total /@ (rstre sezF)) - k sezV)/m
}[[1]];

(*returns list of acceloration vectors of charges. sezV is a list of \
velocities of charges*)

Pol[n_] := {
vmax = 1;
amax = 1;
sezr = Reap[
Do[
Sow[NaključnaTNaKrogli[]]
, n]
][[2]][[1]];
sezv = ConstantArray[{0, 0, 0}, n];
seza = ConstantArray[{0, 0, 0}, n];
t = 1/60;
dt = 1/60;
While[
(Not[
IntegerQ[
t]]) \[Or] ((Max[L[sezv]] > vmax) \[And] (Max[L[seza]] >
amax)),
seza = Seza[sezr, n, 100, 1, 1, sezv];
sezv += seza dt;
sezr += sezv dt;
sezr /= L[sezr];
t += dt;
];
<< TetGenLink;
{coords, incidences} = TetGenConvexHull[sezr];
Show[
Graphics3D[{
EdgeForm[],
FaceForm[Cyan],
GraphicsComplex[coords, Polygon[incidences]]}],
Background -> Black,
Boxed -> False
]
}[[1]];

(*returns graphics of polihedron with charges in end state as \
vertecies*)
Pol[78]

• You don't need to define this l[vektor_] := Sqrt[vektor.vektor]; yourself. Just use Norm[vektor] and also vektor/l[vektor]; just use Normalize[vektor]. For the random point on the sphere, just use RandomPoint[Sphere[{0,0,0},r]]. You're using a strange pattern Reap[Do[Sow... in some places where you could just use a Table. You may also be interested in SpherePoints` – flinty Jun 28 '20 at 0:32
• – flinty Jun 28 '20 at 0:37