The Energy of Spin-Orbit Coupling

Can you show that the energy of interaction is proportional to the scalar product s∙l.? See attachment for symbols.

The energy of a magnetic moment mu in a magnetic field B is equal to their scalar product (see attachment). If the magnetic field arises from the orbital angular momentum of the electron, it is proportional to the l: if the magnetic moment mu is that of the electron spin, then it is proportional to s. It then follows that the energy of the interaction is proportion to the scalar product s-l.

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... center of the sphere contains

If the ring completes a revolution at time T then its angular velocity is and the current in the ring is:
The area of the ring is, so the contribution from a single ring is:
The total moment of the sphere is the sum of all the contributions of all the rings - the integral over the entire volume:
Note that
And if we use where Q is the total charge of the sphere we have:
What is the angular momentum of such a sphere. If the mass of the sphere is then its moment of inertia is
And the angular momentum is therefore:
We see that the ...