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How strong is the magnetic field due to the spin of the electron? Can we derive it from it's magnetic moment?

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If you know the magnetic moment of anything, you can figure out the magnetic field it produces. You can look up the magnetic field of a magnetic dipole with moment $\vec{m}$ in any E&M textbook.

For example in chapter 5 of Introduction to Electrodynamics by Griffiths, you could find the magnetic field at position $\vec{r}$ is $$ \vec{B}_\mathrm{dip}(\vec{r}) = \frac{\mu_0}{4\pi\, r^3}\,\left[ 3(\vec{m}\cdot\hat{r})\hat{r} - \vec{m} \right].$$

The field is proportional to $|m|$ and falls off like $1/r^3$. The details of the directions of things are harder to see from the equation, but Wikipedia has a nice magnetic field line diagram.

To see how the equation gives the picture, try plugging in $\vec{m} = m\, \hat{z}$, for a magnetic moment pointing straight up, and $\vec{r} = z\,\hat{z}$ to find the field on at a point directly above the dipole. Or try a point $\vec{r}$ in the $(x,y)$-plane to see that the field points straight down.

If you look up the spin magnetic moment of an electron you can calculate that: $$B_e \approx \frac{\mu_0 \, m}{4\pi\,r^3} \approx \frac{10^{-30}\, \mathrm{T}\,\mathrm{m}^3}{r^3}$$ So if you are $1$ nm away from an electron the magnetic field is on the order of $1$ mT (milli-tesla).

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The electron is the origin of two fields. Both fields - the magnetic and the electric - are invariant:

  • The elementary magnetic dipole is a constant with the value $-9.28*10^{-24}JT^{-1}$ (NIST).
  • In an unbounded state, the electric elementary charge is a constant with the value $1.60*10^{-19}C$ (NIST).

Electrons are in unity both magnetic dipoles and electric elementary charges..

There is no need to mention spin at all. Any macroscopically detectable magnetic field is simply the sum of the orientations of the magnetic dipoles of the subatomic particles involved.

What prevents us from accepting this simple conclusion is the viewpoint of one hundred years ago, when the magnetic field of the electron was discovered and an intrinsic dual field of the electron simply could not be imagined. Instead, one assumed a rotation, which was quickly discarded again, but the spin had already been put into the world and has tenaciously kept itself alive ever since.

Therefore, the more obvious question would be: How strong is the magnetic field due to the magnetic moments of the aligned subatomic particles?

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