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I have read an electron has an intrinsic magnetic dipole moment.

Does this mean that because dipole moment can be thought of as a current loop, and a current loop radiates EM Waves due to a changing $dJ/dt$ that electrons must radiate EM waves when they are stationary?

can someone show me the math of the electrons dipole moment in maxwells equations? i assume the curl of $M$ is in current density

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The electron doesn't radiate because the "current" in the Amperian model of dipoles is constant. Alternatively, the dipole can be represented by two magnetic monopoles separated by an infinitesimal distance (Gilbert dipole), which also confirms the static picture.

In reality, the electron's magnetic moment is a quantum-mechanical phenomenon which is related to its intrinsic angular momentum (spin) by $$\mathbf{m} = -g\mu_B \mathbf{S},$$ where $\mathbf{S}$ is the spin of the electron, $g=2$ classically, and $\mu_B$ is the Bohr magneton.

As you said, the magnetic moment enters the Maxwell's equations via the source term, and is given by $$\mathbf{J} = -(\mathbf{m}\times \nabla)\delta^{(3)}(\mathbf{r} - \mathbf{r}_0).$$ In media with distributed magnetization, a magnetization density $\mathbf{M}$ is used instead, and the current will need to be integrated over.

Computationally, magnetic contributions from the electron are usually ignored at leading order. This is on one hand due to the massive ratio between the electric and magnetic forces, $$\bigg(\frac{1}{4\pi\epsilon_0}\bigg)\bigg/\bigg(\frac{\mu_0}{4\pi}\bigg)= \frac{1}{\mu_0\epsilon_0} = c^2.$$ It is on the other hand due to the faster decay of the magnetic field of a dipole, being $$\mathbf{B} = \frac{\mu_0}{4\pi}\bigg(\frac{3(\mathbf{m}\cdot\mathbf{r})\mathbf{r}}{|\mathbf{r}|^5} - \frac{\mathbf{m}}{|\mathbf{r}|^3}\bigg) \sim \frac{1}{r^3},$$ compared to the $1/r^2$ decay of the electric field.

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  • $\begingroup$ Thanks for showing me the math, however i have to slightly disagree with your statement that the current in the amperian model is constant. Well... yes the CURRENT is constant however current density changes throughout though? as it changes direction, and experiences a centripital acceleration much like a cyclotron. so why does this not produce radiation? as if you model a current loop using maxwells equations, you DO get radiatiative terms. EDIT: after more research current loops in the real world DO radiate. however aproximating a current as continuous makes the radiative terms dissapear. $\endgroup$ Jul 2 at 9:18
  • $\begingroup$ This seems very strange to me.... almost as if magnetisation on the qauntum scale could perhaps lose this informstion about radiation due to continuos approximations $\endgroup$ Jul 2 at 9:21
  • $\begingroup$ That's a good question. I refer you to this great exposition by McDonald: physics.princeton.edu//~mcdonald/examples/steadycurrent.pdf $\endgroup$
    – jsborne
    Jul 2 at 15:14

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