- #1
Finding potential of a cone using Laplaces Equation
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Homework Help Overview
The discussion revolves around using Laplace's Equation to determine the electric potential of a cone, specifically focusing on the dependence of potential on the angle theta, which is defined as the angle between the positive Z-axis and the surface of the cone.
Discussion Character
- Conceptual clarification, Assumption checking
Approaches and Questions Raised
- Participants explore the reasoning behind the assumption that potential varies only with angle theta, questioning the roles of other variables such as phi and radius. They discuss the implications of the cone being a conducting surface and how equipotential surfaces relate to the geometry of the cone.
Discussion Status
The discussion is active with participants questioning the assumptions made about the potential's dependence on different coordinates. Some have provided insights into the nature of equipotential surfaces and how they relate to the geometry of the cone, but no consensus has been reached regarding the specific dependencies of the potential.
Contextual Notes
There are mentions of the cone being conducting and possibly infinite, as well as references to the behavior of potential in relation to different geometries, but these aspects remain under discussion without definitive conclusions.
- #2
BvU
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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 ?
And: does it say (somewhere on the previous page, for example) that the cone is conducting and infinite ?
- #3
BvU
Science Advisor
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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.
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.
- #4
Miike012
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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
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
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- #5
Miike012
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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.
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.
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