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 [itex]\mathbf{\rho} \cdot (\mathbf{\rho} +\mathbf{n})/ | \mathbf{\rho +n|}^{3} = \nabla_{\rho}(1/|\mathbf{\rho}+\mathbf{n}|), [/itex] the integral [itex] \int {\mathbf{\rho} \cdot (\mathbf{\rho} +\mathbf{n})/ \rho^3 | \mathbf{\rho +n|}^{3}} [/itex] can be easily shown to be equal to to [itex] 4\pi [\itex].

The Attempt at a Solution

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

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

twice to rewrite

[itex]\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]. [/itex]

If we integrate by parts we find

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

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 [itex]1/\rho[/itex] is the Green function for the 3d Laplace equation.