Please answer and label parts a-f. Thank you so much in advance! Adobe Reader To
ID: 1658544 • Letter: P
Question
Please answer and label parts a-f.
Thank you so much in advance!
Adobe Reader Touch 1. Electric field of a hydrogen atom In our study of electrostatics we have considered both point charges and continuous distribu tions of charge. But so far, for simplicity, we have assumed that the continuous distributions have uniform density. In this problem we relax that assumption In the quantum mechanical description of a hydrogen atom the proton is a point charge with charge located at the origin. The electron is a cloud: mybe deled as a continuous charge distribution. The electron cloud is spherically symmetrical and the charge density depends only on distance from the proton. The charge density is given by (r) = exp ( Here a = 0.53A = 5.3 x 10-11 m is the "Bohr radius of the hydrogen atom rtir Figure 2: A proton surrounded by an nvisible electron cloud. Shown in the figure are a shell and a Gaussian surface. a) Consider a spherical shell of inner radius r and outer radius r +dr centered about the proton. What is the charge within this shell? [Answer: -rexp-rfa)/2aExplanation / Answer
given volume charge density
rho(r) = -q*exp(-r/a)/8*pi*a^3
here, a = 5.3*10^-11 m
a. considering a spherical shell of radius r and thickness dr
charge on this shell, dq = rho(r)*4*pi*r^2*dr = -q*exp(-r/a)*r^2*dr/2*a^3
b. for a gaussean surface of radius R
charge enclosed Q = integrate dQ form r = 0 to r = R
Q = integrate(-q*exp(-r/a)*r^2*dr/2*a^3) = (q*(2a^2*R + aR^2)exp(-r/a)/2*a^3)
So adding charge of proton
Q = q + (q*[(2a^2*R + aR^2 + 2a^3)exp(-R/a) - ( 2a^3)] /2*a^3)
hence this is the charge contained in the gasuzsean surface of radius R centered at the origin
c. for r - > infinity
Q = q + (q*[(2a^2*R + aR^2 + 2a^3)exp(-R/a) - ( 2a^3)] /2*a^3)
Q = q - q = 0
d. form symmetry, the electric field is constant in part B surface, and as charge becomes 0 at r -> infinity, this means that for finite r, Q > 0
hence the electric field points radially outwards