# Magnetic field of a spinning electron?

#### kiwiheretic

What sort of magnetic field does a stationary spinning electron have? Is it the same as the magnetic field of a rotating sphere or spherical shell?

#### topsquark

Forum Staff
What sort of magnetic field does a stationary spinning electron have? Is it the same as the magnetic field of a rotating sphere or spherical shell?
Rotation can be a little tricky. No, the electron will have no angular momentum if it is alone and "not moving" (a thing that the Heisenberg Principle does not allow.) In any event the electron has no velocity so it does not feel a magnetic field if the field is constant.

However electrons do have a sort of "built in" angular momentum: We say that an electron has spin 1/2. Spin is inherent to the electron and this never changes under any circumstance. (Spin 1/2 means either spin component hbar/2 or -hbar/2 in this context.) The odd thing here is that this is not a real angular momentum. A "rotation" for the electron is that it has to "spin around" twice to get back to its original orientation. This is purely a Relativistic property and has nothing to do with an actual rotation.

Spin only comes in units of 1/2, 3/2, 5/2,.. (for fermions) or integer spin 0, 1, 2,... (bosons.) For massless particles, like the photon or neutron, we use the term "helicity" to describe the same thing. There is a difference between spin and helicity but for understanding the basic concept this difference is meaningless.

-Dan

Addendum: After re-reading your post you might be thinking about something called the "Bohr magneton." In this we can generally pretend that the electron does generate a magnetic field because it is "spinning." Again, the electron spin is not an angular momentum but in this case we can calculate (via the Dirac equation) the "magnetic field" that spin generates. Like I said, rotation is tricky.

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#### Woody

Is this another of those cases where thinking of the electron in "classical" terms
(as a tiny sphere)
leads to misinterpretations of quantum phenomena?

#### topsquark

Forum Staff
Is this another of those cases where thinking of the electron in "classical" terms
(as a tiny sphere)
leads to misinterpretations of quantum phenomena?
Yes and no. All of my Physics instructors generally talked about it in terms of "rotation," Mathematically speaking, even if they were careful to separate the two. I didn't learn about the Mathematical difference between rotation and spin until about 10 years ago. I wasn't taking classes then. The Math you run into in the undergraduate program simply used the rotation formalism to do calculations. I can't blame them because it is simpler to just treat spin as a rotation in most cases.

-Dan

#### kiwiheretic

Is it just an ugly rumour, then, that an electron, placed in an external magnetic field, lines up it's spin axis to follow the magnetic field lines of the that external field?? They talk about a magnetic dipole moment but how can anyone be so certain that such a thing exists for an electron if there is no rotation or angular momentum?

#### topsquark

Forum Staff
Is it just an ugly rumour, then, that an electron, placed in an external magnetic field, lines up it's spin axis to follow the magnetic field lines of the that external field?? They talk about a magnetic dipole moment but how can anyone be so certain that such a thing exists for an electron if there is no rotation or angular momentum?
To answer your comment on how electron spin can be found in a magnetic field, reference the "Stern-Gerlach experiment." An electron beam passing through a magnetic field will separate into two beams, one with spin 1/2 and the other with spin -1/2.

As I mentioned previously there is something called the Bohr magneton. This is the quanta of the electron magnetic moment. This is due to spin but remember the electron is a point particle... it has no radius to be spinning around. This is another example of spin coming out looking like an angular momentum but isn't.

-Dan