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## Homework Statement

Find the electric charge centred in a sphere of radius a, centered at the origin where the electric potential is found to be (in spherical coordinates) [itex] V(r)=kr^-2 [/itex] where k is some constant.## The Attempt at a Solution

We have [itex] E=-\nabla V = -2kr^{-3} \hat{r} [/itex]

So applying Guass's law to the sphere of radius a, we get

[itex] \oint_s E \cdot da = \frac{Q}{\epsilon_0} [/itex]

And thus [itex] Q= \frac{-8\pi k \epsilon_0}{a} [/itex]

My problem is, surely the same result should be obtained by taking the triple integral of the charge density with respect to volume, but pursuing this path...

[tex] \rho=-\epsilon_0 \nabla^2 V(r) = -\epsilon_0 2kr^{-4} [/tex]

Attempting to integrate this in spherical coordinates results in,

[tex]Q= \iiint_V \rho dV = -8\pi \epsilon_0 k \int_0^a r^{-2} dr [/tex]

but due to the singularity this tends to infinity. Where did I take a wrong step?