- #1
plasmoid
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In the x-y plane, we have the equation
[tex]\nabla^{2} \Psi = - 4\pi \delta(x- x_{0}) \delta (y- y_{0}) [/tex]
with [tex] \Psi = 0 [/tex] at the rectangular boundaries, of size L.
A paper I'm looking at says that for
[tex]R^{2}[/tex] = [tex](x-x_{0})^{2}[/tex] + [tex](y-y_{0})^{2}[/tex] << [tex]L^{2}[/tex] ,
that is, for points very close to the source, the solution must behave as if the boundaries were at infinity, and
[tex] \Psi \approx -2 ln R. [/tex]
I see that -2 ln R will satisfy the equation, but why should it be the only solution valid at R<<L? And how does it satisfy the boundary condition?
- 2 ln R goes to infinity as R goes to zero, does that have anything to do with it?
Morse and Feshbach have the same thing on Pg 798 of vol 1 of their Methods ... , but I can't see an explanation there either.
[tex]\nabla^{2} \Psi = - 4\pi \delta(x- x_{0}) \delta (y- y_{0}) [/tex]
with [tex] \Psi = 0 [/tex] at the rectangular boundaries, of size L.
A paper I'm looking at says that for
[tex]R^{2}[/tex] = [tex](x-x_{0})^{2}[/tex] + [tex](y-y_{0})^{2}[/tex] << [tex]L^{2}[/tex] ,
that is, for points very close to the source, the solution must behave as if the boundaries were at infinity, and
[tex] \Psi \approx -2 ln R. [/tex]
I see that -2 ln R will satisfy the equation, but why should it be the only solution valid at R<<L? And how does it satisfy the boundary condition?
- 2 ln R goes to infinity as R goes to zero, does that have anything to do with it?
Morse and Feshbach have the same thing on Pg 798 of vol 1 of their Methods ... , but I can't see an explanation there either.
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