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If the total charge is non-zero, the dipole moment is ill defined in that its value depends on the choice of origin. For total charge zero distributions, the dipole moment does not depend on choice …

The electric dipole moment of a charge distribution about a point ${\bf R}$ is independent of ${\bf R}$ only when the total charge is zero. In that case $$ p({\bf R})\equiv \int ({\bf r}-{\bf R}) \r …

There is no tangential force. Both forces due to the central charge on the circumferential charges are exactly radial. So they have no effect on moving the charges on the ring.

I think that the correct expression involves $\delta'(k-k')$ rather than a partial derivative. Suppose we construct a wave-packet with states taken from a single band with index $n$: $$ \varphi({\bf …

I think that rather muddied some of the issues in my original question about electron electric dipole moments --- so here are some comments, and perhaps a further implied question, about about the Lo …

Quantum mechanically, you would calculate the dipole moment of the atom by using perturbation theory to computing the change in the ground state energy under the introduction of a small magnetic field …

L&L intend the integral to be over the entirety of 3-space, not only the interior of the polarized body. The surface integal is zero because ${\bf P}$ is zero at infinity

The magentic dipole moment is part of a skew symmetric Lorentz tensor $M_{\mu\nu}$ which is defined so that the interaction with an electromagentic field is $\propto M_{\mu\nu}F^{\mu\nu}$. What is …