Electrostatic interaction energy example (jackson)

[rafaelpol]

Homework Statement

I am trying to follow a derivation in Jackson - Classical Electrodynamics

Homework Equations

In equation 1.58 (2nd/3rd edition) of Jackson - Classical Electrodynamics he says that by using the fact that $\mathbf{\rho} \cdot (\mathbf{\rho} +\mathbf{n})/ | \mathbf{\rho +n|}^{3} = \nabla_{\rho}(1/|\mathbf{\rho}+\mathbf{n}|),$ the integral $\int {\mathbf{\rho} \cdot (\mathbf{\rho} +\mathbf{n})/ \rho^3 | \mathbf{\rho +n|}^{3}}$ can be easily shown to be equal to to $4\pi [\itex]. <br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> I can't really follow on how to solve this integral once the fact mentioned above is known. I know how to solve the integral using spherical coordinates, but from what I have seen that does not follow from what Jackson said at all. I am just curious if there is an easier to evaluate the integral using the gradient identity.$

 

[fzero]

You have a typo, but that's probably not the problem. We actually have to use the identity

$\frac{\vec{\rho}+\hat{n}}{ | \vec{\rho}+\hat{n}|^3 } = - \nabla_\rho \left( \frac{1}{ | \vec{\rho}+\hat{n}|} \right)$

twice to rewrite

$\int d^3\rho \frac{ \vec{\rho}\cdot ( \vec{\rho}+\hat{n}) }{ \rho^3 | \vec{\rho}+\hat{n}|^3} = \int d^3\rho \left[ \nabla_\rho \left( \frac{1}{ \rho} \right) \right] \cdot \left[ \nabla_\rho \left( \frac{1}{ | \vec{\rho}+\hat{n}|} \right) \right].$

If we integrate by parts we find

$\int d^3\rho \nabla_\rho\cdot \left[ \frac{1}{ | \vec{\rho}+\hat{n}|} \nabla_\rho \left( \frac{1}{ \rho} \right) \right] <br /> -\int d^3\rho \frac{1}{ | \vec{\rho}+\hat{n}|} \nabla_\rho^2 \left( \frac{1}{ \rho} \right) .$

We can use the divergence theorem to show that the first, total derivative, term vanishes, while for the 2nd term, we use the fact that $1/\rho$ is the Green function for the 3d Laplace equation.