The hydrogen atom in its ground state can be modeled as a positive point charge

Question

The hydrogen atom in its ground state can be modeled as a positive point charge of magnitude +e (the proton) surrounded by a negative charge distribution that has a charge density (the electron) that varies with the distance from the center of the proton r as ?(r) = ?0e-2r/a (a result obtained from quantum mechanics), where a = 0.0529 nm is the most probable distance of the electron from the proton.

(a) Calculate the value of ?0 needed for the hydrogen atom to be neutral.

?0 = _________ C/m3

(b) Calculate the electrostatic potential (relative to infinity) of this system as a function of the distance r from the proton. (Use the following as necessary: a, e for the electron's charge, k, and r.)

V = __________

Explanation / Answer

charge density is the charge per unit volume. to be neutral means the total charge has to be -e. integrate the charge density over volume to get the total charge. in polar coordinates this is
-e = integ(0,inf) [ rho0 exp(-2r/a) *4*PI*r^2 * dr ]
you integrate by parts twice to do the integral

find the amount of charge inside a sphere of radius r by integrating up to r and adding +e, this is q(r)
then V(r)=k q(r)/r

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