Electron dipole moment: Any electrically charged rotating object, will have mech

Question

Electron dipole moment: Any electrically charged rotating object, will have mechanical angular momentum L and a magnetic dipole moment m in equal or opposite directions (equal if the charge is positive, opposite if the charge is negative). The ratio between the magnetic moment and the anguar momentum is called "gyromagnetic ratio" Y.

a) Prove that for a solid charged sphere with uniform mass and charge density, and total mass M and total charge Q, rotation at some angular velocity w, the dipole magnetic moment m and the angular momentum L are related as m=(Q/2M)L.

b) Remembering the angular momentum is quantized in multiples of h-bar, we write the electron's spin magnetic dipole momentum as u = - guBL/h-bar, where uB = e(h-bar)/2me is the "Bohr magneton", and g is a dimensionless factor. Considering the electron as a rotating sphere with uniform charge and mass density, get an estimate for the electorn's "g-factor" ge, and compare it with its CODATA measured value, from https://physics.mist.gov

c) What angular velcotiy w would produce a magnetic dipole moment uB?

d) Consider the electron not as a uniformly charged sphere, but with a uniform mass density and a charge density proportional to e-2r/a. What is the new estimate for ge?

e) Consider the classical Bohr model of the hydrogen atom, with an electron in a circular orbit around the nucleus, and the possible orbits having a quantized (orbital) angluar momentum L=nh. Consider an effective electrical current due to the electron motion in terms of the electron charge and angular frequency, and find an expression for the orbital magnetic dipole moment uL in terms of the angular momentum L. What is the orbital g-factor gL such that uL = -gLuBL/(h-bar)?

Explanation / Answer

a. for a solid charged sphere

I = 2mr^2/5

now, for total charge Q

charge density = 3Q/4*pi*r^3

at radius r from the center

current = di

di = rho*2*pi*r*dr*2*sqroot(R^2 - r^2)*w/2*pi

hence dm = di*pi*r^2

hence m = integral dm = integral rho*2pi*r^3*dr*sqroot(R^2 - r^2)*w from r = 0 to r = R

hence

m/L = Q/2M

b. u = -guBL/hbar

uB = e^(hbar)/2me

hence gyromagnetic ratio

m/L = Q/2m = 87912087912.08

c. for uB

uB = (Q/2m)*2*m*r^2*w/5

w = 5uB/Qr^2

where r is radius of electron

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