Electric Field Inside a Hydrogen Atom

[viperassasin]

Homework Statement

A hydrogen atom is made up of a proton of charge + Q=1.60 \times 10^{ - 19}\; {\rm C} and an electron of charge - Q= - 1.60 \times 10^{ - 19}\; {\rm C}. The proton may be regarded as a point charge at r=0, the center of the atom. The motion of the electron causes its charge to be "smeared out" into a spherical distribution around the proton, so that the electron is equivalent to a charge per unit volume of \rho (r)= - {\frac{Q}{ \pi a_{0} ^{3}}} e^{ - 2r/a_{0}} where a_0=5.29 \times 10^{ - 11} {\rm m} is called the Bohr radius

Find the total amount of the hydrogen atom's charge that is enclosed within a sphere with radius r centered on the proton.

Homework Equations

The Attempt at a Solution

do I just try and divide out the volume.

 

[nicksauce]

Well one can presumably just integrate charge density from 0 to r to get charge enclosed.

 

[tony873004]

Here's my guess:
between the proton and the electron, the electron's charge cancels out, since there is no charge inside a uniformly charged sphere. So if r is less than the electron's distance than charge = +Q. If r is greater than the electron's distance, outside a uniformly charged sphere, the sphere can be treated as a point mass at the center of the sphere. So outside the electron's position charge = +Q-Q = 0.

 

[viperassasin]

http://session.masteringphysics.com/render?infix=Q*(1+2*r/a_0+2*(r/a_0)^2)*e^(-2*r/a_0)

This the answer