Finding potential of a cone using Laplaces Equation

[Miike012]

In the example in the attachment, Laplaces Equation is used to find the potential of a cone.

My qustion is, How do they know the potential only varies with angle theta (theta is the angle between the positive Z-axis and the surface of the cone.)

 

[BvU]

Good question. After all, in the gap $\theta=0$ and the potential goes from 0 to V₀ over a (very short?) distance !
And: does it say (somewhere on the previous page, for example) that the cone is conducting and infinite ?

 

[BvU]

Coming back to your question: You are inclined to believe V does not depend on $\phi$ ? If so, why / why not ?

The only other spherical coordinate I can think of is r. Look at the picture and compare a picture of a point with r = 5 to a picture of a point with same theta and with r = 1, but that pictre is five times enlarged.

 

[Miike012]

I know the E-Filed at a point on the surface of the cone is perpendicular to the surface of the cone. Therefore the surface of the cone is an equipotential surface.

In spherical coordinates we have (R,θ,ø)
R is the distance from the origin
θ is the angle from the positive z-axis
ø is the angle from the positive x-axis

Looking at the picture in the attachment you can see as R changes (The orange line) while ø and θ are constant, The potential does not change

Now changing ø while keeping R and θ constant, V does not change

Now changing θ while keeping R and ø constant, V changes.. Assuming that the potential on the surface of the blue cone is different from the surface of the green cone.

Thats the best explanation I can come up with

 

[Miike012]

Just another though,
considering the surface of a conductor, we know the equipotential surfaces very neer the surface of the conductor is neerly the shape of that conductor.
Therefore, by knowing the equation of the surface we also know the equation of the equapotential lines neer the surface of the conductor.
(Getting to my point very soon)
Take a Conducting Sphere for example, We can represent the sphere in spherical coordinates by R, if I am correct I believe V varies by R.
Take a parallel plate capacitor, two plates parallel to the xy plane, one plate in the plane Z = 0 and Z = zi. essentially you can think of the surface of one of the capacitors as a constant function of Z.. and varies in the Z direction..
NOW the cone. A distinc surface of a cone is defined by keeping the angle θ between the +z-axis constant, hence V for distinct cones vary by θ in spherical coord.