I Where is Laplace's Equation Valid in Different Domains?

AI Thread Summary
Laplace's equation is valid in regions defined by specific boundary conditions, which must be specified for accurate solutions. The discussion highlights the importance of understanding whether the solutions derived from the equation apply only within the boundaries or can extend beyond them. It raises questions about the validity of solutions when potential is specified on conductors and the implications for static fields. The distinction between solving Laplace's equation and Poisson's equation is also emphasized, particularly in relation to charge densities at boundaries. Ultimately, the validity of the equation depends on the convergence of the series and the relationship between fields inside and outside the defined region.
yucheng
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Hi!

This thread might well be similar to:
https://www.physicsforums.com/threa...classical-electrodynamics-3rd-edition.910410/

I'm self-studying Vanderlinde and having a great time. However, I think that I am conflating and confusing many different things. Let me just ask them here:

Chapter 5 Laplace's equations.
  1. When solving the equation, ending in a series expansion, in what domain is the expansion valid?

    1638353977752.png

    The potential is specified for side 4, while the other sides are set to zero. From what I know of, the potential is a harmonic function i.e. it takes on a maximum or a minimum at the boundaries. Does this mean that the solution I have found is only valid within the boundary? I think so.

    But then, you can specify two disconnected, nested boundaries like the circle as the lower bound to the radial distance, and infinity.

    So, the author arrives at the equation

    1638354203530.png


    So, where is equation 5-15 valid? It can be used for both the region within and the region outside of the parallelepiped right, as long as the boundary conditions are specified (and delineates the bounding region)?
  2. On page 93, Vanderlinde mentions that we are solving for static fields with charges on the boundary. For problems where a potential is specified on a conductor, find the potential in space, fields etc., does this mean that the equation is only valid for the region of interest, but not the conductor?
  3. In Example 5.3 on page 100, the author gives coaxial nonconducting cylinders with surface charge densities giving rise to potential, and solves the Laplace's equation by specializing V with r = a.
    1638355150214.png

    But doesn't this mean that ##\nabla^2 V \neq 0 ## anymore, at least on the boundary? So... why are we not solving Poisson's equation instead, with a dirac delta function for the charge density, of course? :smile:
  4. Why is ##\ln r## needed to to have a nonzero net charge?
1638355402908.png

Thanks in advance!
 
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yucheng said:
So, where is equation 5-15 valid?
I think this depends on what one means by valid. May this expression be numerically evaluated outside the region for which it was initially derived versus does it still represent the actual field outside the region. If the series converges outside the region then the answer is maybe but usually not. For example a closed metal box the fields outside and inside are generally unrelated.
 
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