# Potential of a solid, double cone shaped charge distribution

• Philip Koeck
In summary, the conversation discusses the possibility of finding an analytical expression for the electrostatic potential produced by a charge distribution confined to a double cone shaped region. The problem involves computing the potential at a given point created by a disk of radius r at a certain ordinate along a fixed axis. Various approaches are suggested, including using elliptic integrals and integrating over different rings and z values, as well as considering a point crossing the two cones and computing the potential generated by lines in spherical coordinates. The complexity of the problem is acknowledged and it is suggested to consult a specific resource for more information.
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.

Delta2
You've forgotten to precise at which points you need the potential.

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
coquelicot said:
You've forgotten to precise at which points you need the potential.
Everywhere. Both inside and outside the charge distribution.

Philip Koeck said:
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.

Philip Koeck
coquelicot said:
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.

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.

Philip Koeck

## 1. What is the potential of a solid, double cone shaped charge distribution?

The potential of a solid, double cone shaped charge distribution refers to the electric potential energy per unit charge at a given point in space, due to the arrangement of positive and negative charges in a solid, double cone shape.

## 2. How is the potential of a solid, double cone shaped charge distribution calculated?

The potential of a solid, double cone shaped charge distribution can be calculated using the formula V = k(Q1/r1 + Q2/r2), where V is the potential, k is the Coulomb's constant, Q1 and Q2 are the charges on the two cones, and r1 and r2 are the distances from the point to each cone.

## 3. What factors affect the potential of a solid, double cone shaped charge distribution?

The potential of a solid, double cone shaped charge distribution is affected by the magnitude and distribution of the charges on the cones, as well as the distance from the point to each cone. Additionally, the direction of the electric field lines and the dielectric constant of the surrounding medium can also impact the potential.

## 4. How does the potential of a solid, double cone shaped charge distribution vary with distance?

The potential of a solid, double cone shaped charge distribution follows an inverse relationship with distance. This means that as the distance from the point to the cones increases, the potential decreases.

## 5. What are some real-world applications of the potential of a solid, double cone shaped charge distribution?

The potential of a solid, double cone shaped charge distribution has various applications in fields such as physics, engineering, and electronics. For example, it can be used to study the electric potential of charged particles in a solid, double cone shaped material, or to design and optimize electronic devices such as capacitors and transistors.

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