Consider an electron in circular motion around a cirlce of radius r. a) Find an
ID: 2065469 • Letter: C
Question
Consider an electron in circular motion around a cirlce of radius r.
a) Find an equation for the electron's magnetic dipole moment, u, for this circular motion in terms of the electron's angular momentum L. HINT: For a point particle, the angular momentum is L = r x p where p = Mv is the (linear) momentum of the electron.
b) A simple model for magnetism due to spin, simply replaces the angular momentum, L, from the previous expression with the spin angular momentum S. An excellent result of the more fundamental theory of motion, quantum mechanics, is that the z-component of an electron's spin angular momentum, Sz, can have only one of two values Sz=+or- 1/2h, where h = 1.06 x 10^-34J.s is a constant (called the reduce Plank's constant). Using your expression from part a, find the z-component of the electron's spin magnetic dipole moment, (Uz)s' in our "simple" model for the case when Sz=+1/2h.
c)In a real experiment, when an electron is placed in a magnetic field of 1.5T, its potential energy will increase by 2.8 x 10^-23J when its spin flips from being opposite magnetic field to being parrallel to it. What is the real value of an electron's spin magnetic dipole moment, (Uz)R?
Explanation / Answer
a)
the angular momentum is:
L = r x p
= m (r x v)
= m r v sin
= m r v sin90
= m r v
>>>> v r = L/m -----> equation (1)
i = dq/dt
= e/(2r/v)
= ev/(2r)
= i A
= ev/(2r) * ( r2)
= e v r / 2
from (1) >>> = eL/(2m)
(e = 1.6*10^-19 C)
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b)
Sz=+1/2h = 0.5 * 1.06 x 10^-34 = 0.53 x 10^-34
= eL/(2m) = 1.6e-19*0.53e-34/(2*9.109e-31) = 4.655e-24
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c)
U = ( B cos0) - ( B cos(180))
= 2 B
>>> = U/2B = 2.8e-23/(2*1.5) = 9.33e-24