- #1
Buffu
- 849
- 146
I want to prove ##\displaystyle U = {1\over 8\pi}\int \vec E \cdot \vec E dV## and ##\displaystyle U = \frac12 \int \phi \rho dV## are equal.
I started with ##\nabla \cdot (\phi \nabla\phi) =(\nabla \phi)^2 + \phi \nabla^2 \phi##'
Then
##\displaystyle {1\over 8\pi}\int \vec E \cdot \vec E dV = {1\over 8\pi}\int \nabla \phi \cdot \nabla \phi dV = {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) - \phi \nabla^2 \phi dV = {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) dV - {1\over 8\pi}\int \phi \nabla^2 \phi dV ##
By poisson's equation, ##\nabla^2 \phi = 4\pi \rho##
So, ##\displaystyle {1\over 2}\int \rho \phi dV##
So I guess I need to prove ##\displaystyle {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) dV## is zero
Using divergence theorem,
## \displaystyle {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) dV = {1\over 8\pi}\int (\phi \nabla \phi)\cdot d\vec a##
Now I don't have any clue what to do, I guess I need to use ##-\nabla \phi = \vec E## but I don't know how.
I started with ##\nabla \cdot (\phi \nabla\phi) =(\nabla \phi)^2 + \phi \nabla^2 \phi##'
Then
##\displaystyle {1\over 8\pi}\int \vec E \cdot \vec E dV = {1\over 8\pi}\int \nabla \phi \cdot \nabla \phi dV = {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) - \phi \nabla^2 \phi dV = {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) dV - {1\over 8\pi}\int \phi \nabla^2 \phi dV ##
By poisson's equation, ##\nabla^2 \phi = 4\pi \rho##
So, ##\displaystyle {1\over 2}\int \rho \phi dV##
So I guess I need to prove ##\displaystyle {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) dV## is zero
Using divergence theorem,
## \displaystyle {1\over 8\pi}\int \nabla \cdot (\phi \nabla\phi) dV = {1\over 8\pi}\int (\phi \nabla \phi)\cdot d\vec a##
Now I don't have any clue what to do, I guess I need to use ##-\nabla \phi = \vec E## but I don't know how.
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