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1. Homework Statement

1. Homework Statement

Calculate the work needed to move a point-charge with the charge q from infinity to the center of the chargedistribution given by:

[itex]\rho(\vec{r})=\rho_0 e^{-\alpha r}[/itex]

## Homework Equations

[itex]U=qV[/itex]

[itex]V=\frac{1}{4\pi\epsilon_0}\int_{V}\frac{\rho(\vec{r})}{r}dV[/itex]

## The Attempt at a Solution

I did the integral by using integration by parts:

[itex]\frac{1}{4\pi\epsilon_0}\int_{V}\frac{\rho(\vec{r})}{r}dV=\frac{1}{4\pi\epsilon_0}\int_0^{\infty}\int_0^{2\pi}\int_0^{\pi}\frac{\rho_0 e^{-\alpha r}}{r}r^2=\frac{\rho_0}{\epsilon_0 \alpha^2}[/itex]

The work should then be, for moving the charge from infinity to 0, [itex]U=\frac{\rho_0 q}{\epsilon_0 \alpha^2} [/itex]

Is this correct? Please let me know if I should elaborate some steps.