Magnetic Moment, Electron Spin, Energy

[roeb]

Homework Statement

The PE of a magnetic moment in an external magnetic field is given by U = -u dot B. Calculate the difference in energy between the two possible orientations of an electron in a magnetic field B = .6k T

Homework Equations

The Attempt at a Solution

I realize electrons can have an up spin and down spin, but I am having a hard time doing this problem.

I believe I should be using g = 2 for the gyromagnetic ratio. So: (_h = h bar)
u_z = -g u_B J_Z/_h
u_B = e(_h)/(2m_e)

Unfortunately then I don't know what I should have for J_z. (This is defined by my book as "any type of angular momentum).

So I guess my problem is that I don't know what I should use for angular momentum, since n and l aren't defined in the problem. Can anyone help clear this up?

 

[borgwal]

Use the electron spin, S_z. Strange you would know about the gyromagnetic ratio but not about which angular momentum that refers to in the case of an electron!

 

[thomate1]

You are correct that the energy of a magnetic moment in an external magnetic field is given by U = -u dot B, where u is the magnetic moment and B is the magnetic field. In this case, we are dealing with an electron, which has a spin angular momentum of 1/2. This means that the possible values for Jz are +1/2 and -1/2.

Using the equation you provided, uz = -g uB JZ/_h, we can calculate the energy difference between the two possible orientations of the electron in the magnetic field B = 0.6 kT. Since Jz can take on two values, we will calculate the energy difference for both cases:

For Jz = +1/2:
uz = -g uB (1/2)/_h = -g uB/2_h
Plugging in the values for g, uB, and _h, we get:
uz = -(2)(1.602 x 10^-19 C)(0.6 x 10^-3 T)/2(6.626 x 10^-34 J s) = -2.404 x 10^-24 J

For Jz = -1/2:
uz = -g uB (-1/2)/_h = g uB/2_h
Plugging in the values for g, uB, and _h, we get:
uz = (2)(1.602 x 10^-19 C)(0.6 x 10^-3 T)/2(6.626 x 10^-34 J s) = 2.404 x 10^-24 J

Therefore, the energy difference between the two possible orientations of the electron in the magnetic field B = 0.6 kT is 4.808 x 10^-24 J. This represents the difference in energy between the electron's up spin and down spin states. I hope this helps clarify the problem for you.