- #1
demonelite123
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i'm trying to understand how the electrosatic potential expressed as an integral satisfies poisson's equation. i know that i have to take the laplacian of both sides of (Eq 1.17) page 35 in Jackson.
i understood how jackson took the laplacian of [itex]\frac{1}{\sqrt{r^2 + a^2}}[/itex] but after Eq 1.30 i am completely lost. he took a taylor expansion of p(x') around the point x' = x which i understand to second order is p(x) + (x' - x) * ∇p + (1/2)((x' - x) * ∇)2p.
but i have no idea how he calculated the integral over [itex]\theta[/itex] and [itex]\phi[/itex] to get the answer on the next line.
how would one go about this? is the process straightforward enough that jackson chooses to simply omit it in the text?
i understood how jackson took the laplacian of [itex]\frac{1}{\sqrt{r^2 + a^2}}[/itex] but after Eq 1.30 i am completely lost. he took a taylor expansion of p(x') around the point x' = x which i understand to second order is p(x) + (x' - x) * ∇p + (1/2)((x' - x) * ∇)2p.
but i have no idea how he calculated the integral over [itex]\theta[/itex] and [itex]\phi[/itex] to get the answer on the next line.
how would one go about this? is the process straightforward enough that jackson chooses to simply omit it in the text?