Potential inside NON-Conducting hollow spherical shell

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Discussion Overview

The discussion revolves around determining the constants in the expression for the electric potential inside a non-conducting hollow spherical shell with a given surface charge density. Participants explore the implications of symmetry and boundary conditions in the context of electrostatics, particularly focusing on the use of spherical coordinates and the Laplacian operator.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents the potential inside the shell as \( V_{(x,y,z)} = \frac{V0}{R^{2}}(6z^2+ax^2+by^2) \) and seeks to determine the constants \( a \) and \( b \).
  • Another participant suggests using the Laplacian of the potential, stating that \( \nabla^2 V = -\frac{\sigma}{\epsilon} \) on the surface of the shell, and recommends analyzing the problem in spherical coordinates.
  • A participant questions whether solving Poisson's equation on the surface would help in determining \( a \) and \( b \), expressing confusion over the boundary conditions.
  • One participant proposes that if the potential is axially symmetric, then \( b \) must equal \( a \), and notes that the angular distribution should correspond to a Legendre polynomial, specifically \( P_2(\cos\theta) \).
  • This participant concludes that given the power of \( x, y, \) and \( z \) is 2, it follows that \( a = b = -3 \).

Areas of Agreement / Disagreement

Participants express differing views on the approach to determining the constants \( a \) and \( b \). While some suggest using boundary conditions and the Laplacian, others question the clarity of these conditions and the implications of symmetry. The discussion does not reach a consensus on the method for determining the constants.

Contextual Notes

Participants mention the need for boundary conditions and the use of spherical coordinates, but there are unresolved aspects regarding the application of Poisson's equation and the interpretation of symmetry in the potential.

dikmikkel
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Hi Guys,
Suppose we have a spherical shell with charge density on the surface \sigma and radius R. The potential inside the shell is given by:

V_(x,y,z) = \frac{V0}{R^{2}}(6z^2+ax^2+by^2)
It is assumed, that the potential is rotational symmetric around the z-axis inside and outside the shell, and goes to 0 far away from the shell. There's no charge inside and outside the shell and no outer field.

How do i determine the constants a and b?

Mabye change to spherical coordinates and solve the equation:
\frac{\partial{V}}{\partial{\theta}}=0
for a or b. But i can't figure out any other conditions if this is right.
 
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The Laplacian of the voltage is related to the charge. So your boundary conditions would be:

\nabla^2 V = -\frac{\sigma}{\epsilon}

On the surface of the shell. So for ease of analysis you would want to do this in spherical coordinates.
 
I can't really see that this is a boundary condition? And would solving possions equation on the surface help me determine a and b, i can't really follow?
 
Aah sorry I am really tired now. I take the laplacian, and this would relate it all, beautiful, Thanks. Just so used to try finding the potential :)
And sorry for the language. I'm from Denmark.
 
If the pot inside is axially symjmetric, and b must be equal.
Since DEl^2 V=0 inside, the angular distribution must be a Legendre polynomial.
Since the power of x,y,and z is 2, it must be
P_2(\cos\theta)=(3\cos^2\theta-1)/2.
This means a=b=-3.
 
Last edited by a moderator:
That's gold thank you.
 

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