A hydrogen atom can be considered as having a central point-like proton of posit

Question

A hydrogen atom can be considered as having a central point-like proton of positive charge +e and an electron of negative charge -e that is treated as a volume charge distribution around the proton which is given by: p = A * e - 2r / ao and extends out to infinity. A is a constant, a0 is the Bohr radius, and r is the distance from the center of the atom, (a) Using the fact that the hydrogen atom is electrically neutral (meaning that the electron charge distribution must integrate to a charge of -e). find A. (b) Then, find the electric field produced by the atom at the r = a0.

Explanation / Answer

The volume charge density must be integrated with respect to r from infinity to zero, and equated to the charge of the electon. Then solve for A.

-e = [+inf to 0] p(r) a(r) dr, where a(r) is the area of a sphere at radius r.
= [+inf to 0] Aexp( -2r/a0) 4r^2 dr
= A4 [+inf to 0] exp( -2r/a0) r^2 dr

Observe that the above is in the given form of integral [ x^n e^-ax dx ] from infinity to 0 = n!/a^(n +1):
where a = 2/a0, n=2, and x = r
-e = A4(n!/a^(n +1))
= A4(2!/(2/a0)^3)
= A8/(8/a0^3)
= Aa0^3

Solve for A:
A = -e/a0^3

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