- #1

Pengwuino

Gold Member

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So this may be a bit silly, but one thing I've never really learned in all my years is how one actually goes about calculating equipotential surfaces for arbitrary potentials? Let's say I have a potential that goes like

[tex]\Phi = A_0e^{-\left({{r}\over{r_0}}\right)^2}\cos^2 (\phi) \sin^2(\theta)[/tex]

where [itex] r, \phi[/itex] are your typical spherical coordinates and [itex]A_0[/itex] is a constant and [itex]r_0[/itex] is there for dimensional purposes and I want to determine even a single equipotential surface, how does one go about doing this? You typically see very simple examples in physics and you're typically finding something trivial like the shells you see around point particles, but this relies on plain ol intuition. How does one determine the surface with arbitrary (but well-behaved) functions such as my example?

Edit: I changed my function a few times so that it's Laplacian has no weird features for what I'm doing.

[tex]\Phi = A_0e^{-\left({{r}\over{r_0}}\right)^2}\cos^2 (\phi) \sin^2(\theta)[/tex]

where [itex] r, \phi[/itex] are your typical spherical coordinates and [itex]A_0[/itex] is a constant and [itex]r_0[/itex] is there for dimensional purposes and I want to determine even a single equipotential surface, how does one go about doing this? You typically see very simple examples in physics and you're typically finding something trivial like the shells you see around point particles, but this relies on plain ol intuition. How does one determine the surface with arbitrary (but well-behaved) functions such as my example?

Edit: I changed my function a few times so that it's Laplacian has no weird features for what I'm doing.

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