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## Homework Statement

Given the electric potential ##V(r)=A\frac{e^{-\lambda r}}{r}## calculate the charge density ##\rho(r)## and the electric field ##E(r)##.

They specify the answer for charge density should be: ##\rho = \epsilon_0 A(4\pi \delta^3(r)-\lambda^2e^{-\lambda r}/r)##

## Homework Equations

##E=-\nabla V##

##\nabla E = \frac{\rho}{\epsilon_o}##

## The Attempt at a Solution

For the electric field I got: ##E = Ae^{-\lambda r}(1+\lambda r)/r^2##, pointing in the r direction and this is the same as their answer. Then I tried to take the divergence of E, and in spherical coordinates this would be ##\nabla E = \frac{1}{r^2}\frac{\partial r^2 E}{\partial r} = \frac{1}{r^2}\frac{\partial (Ae^{-\lambda r}(1+\lambda r))}{\partial r} ## . However this doesn't give me a delta function so I am not sure where I did something wrong.