Potential of a solid, double cone shaped charge distribution

[Philip Koeck]

Does anybody know if there is an analytical expression for the electrostatic potential produced by a charge distribution confined to a double cone shaped region. Think of a beam of charged particles converging to a focus and then diverging again. The total charge in each thin, cross-sectional slab of this beam has to be constant. Therefore the charge density is inversely proportional to the local, cross-sectional area of the beam. The beam (double cone) can be treated as extending infinitely far in both directions.
Something similar might also help.

 

[coquelicot]

You've forgotten to precise at which points you need the potential.

 

[Gordianus]

If you seek an analytical solution I suggest checking Smythe "Static and Dynamic Electricity". The number of cases it deals with is very large (if the solution isn't there I can't imagine other book)

 

[Philip Koeck]

You've forgotten to precise at which points you need the potential.

Everywhere. Both inside and outside the charge distribution.

 

[coquelicot]

Does anybody know if there is an analytical expression for the electrostatic potential produced by a charge distribution confined to a double cone shaped region. Think of a beam of charged particles converging to a focus and then diverging again. The total charge in each thin, cross-sectional slab of this beam has to be constant. Therefore the charge density is inversely proportional to the local, cross-sectional area of the beam. The beam (double cone) can be treated as extending infinitely far in both directions.
Something similar might also help.

In general the problem amount to computing the potential V(t, r) at a point (x,y,z), created by a disk of radius r at ordinate t along a given axis (the axis is fixed), and uniformly charged with a total charge q. Once you have it, you sum $\int_{-\infty}^{+\infty} V(|\alpha t|) dt$, where $\alpha$ is the coefficient giving the aperture of the cones.
Hope this will help.

 

[coquelicot]

Added: this article seems to suggest this is far from being a simple problem, even for the case of a simple disk.https://www.sciencedirect.com/science/article/pii/S0893965911002564
https://www.sciencedirect.com/science/article/pii/S0893965911002564

 

[Philip Koeck]

Added: this article seems to suggest this is far from being a simple problem, even for the case of a simple disk.
https://www.sciencedirect.com/science/article/pii/S0893965911002564

Yes, it seems so. One way to do it would be to take the potential of a ring, which involves an elliptic integral, then integrate over such rings with different radii to get a disk, and then integrate over z as you described earlier. The integrals are probably nightmares.

 

[coquelicot]

You may try another attack: Let O be the point of crossing of the two cones. Let M be a point in the space, and L be a straight line passing through O.
W.l.g, O is the origin and the axis of the two cones is the x-axis. Now, w.l.g, M is in the plane x-y.
Let d be the distance of a point x on L to M, that is $d = h \cot(\alpha)$ where h is the orthogonal distance of M to d. It may be possible to compute $\int_{-\infty}^{+\infty} \rho(\alpha) d\ d\alpha$, that is, the potential created by L, while $\rho$ is the linear density of charge inherited from the disk of the double cone it is crossing (compute it). Then you have to sum in spherical coordinates the potential generated by the lines L you've just found (). Definitely not trivial.