How can you use the Delta Function to find the Electric field inside of a Hydrogen Atom?

Question

This is a problem from my homework, but I'm lost conceptually on how to do it. I'm not looking for someone to do the work for me, I'm looking for some guidance on how I can set it up/understand it

The Charge density of an electron's cloud in a Hydrogen atom is

$$\rho = - \frac{e}{\pi a^3} \exp(-2 r / a) \, .$$

Where $a$ is the Bohr radius, $r$ is the distance to the proton and the proton has a charge of $+e$. Investigate $E$ at small $r \ll a$ and large $r \gg a$ distances.

In addition to how the question is worded, I also know that we are supposed to use the delta function in our solution, though I'm a tad confused on how to do that.

Thinking about it logically for a moment though, it appears to me that $r \ll a$ should result in a positive electric field, while $r \gg a$ should result in a net $E=0$. Is this assumption correct?

Answer 1

The most simple way of solving this is using Gauss' law and spherical symmetry. Then $$\int_{\delta \Omega} \mathbf{E}\; \text{d}\mathbf{F} = \int_{ \Omega} \nabla \mathbf{E} \; \text{d}V = \int_{ \Omega} \frac{\rho}{\varepsilon_0} \; \text{d}V $$ In the case of spherical symmetry this can be simplified to $$\int_{ \Omega} \mathbf{E}\; \text{d}\mathbf{F} = 4\pi r^2 E_0(r) = 4\pi \int \limits_0^r r'^2\frac{\rho(r')}{\varepsilon_0} \; \text{d}r' $$