Mechanical properties of electric point dipoles
is there a sensible way to introduce a point electric dipole with well defined electrical and mechanical properties?
Partly. We have a few options:
Finite moment of inertia: For finite $d$, the electric dipole moment is $qd$ and the moment of inertia is proportional to $md^2$. We can take $d\to 0$ with both $qd$ and $md^2$ held fixed, but this requires $q\to\infty$ and $m\to\infty$. Taking $q\to\infty$ is not a problem, because the two point-charges have opposite signs, so the net charge is zero. But the two point-masses have the same sign, so the net mass goes to infinity.
Finite mass: Alternatively, we could take $d\to 0$ with $qd$ and $m$ held fixed. Then we would get an electric dipole with finite mass but with zero moment of inertia.
A fully pointlike object can have either a non-zero moment of inertia or a finite mass, but not both. But we also have another option:
- Hybrid model: We can treat an object as pointlike for some purpose and as non-pointlike for other purposes. In a model, there is no reason why we need to keep the charges co-located with the masses. We can use a model of a rigid object with two charges separated by a distance $d_1$ and two masses separated by a distance $d_2$. We can take $d_1\to 0$ while keeping $d_2$ small-but-not-zero.
Which of these three options we should use depends on what we're trying to accomplish.
can we conclude that point electric dipoles are not consistent constructs in physics?
Consistent with what?
...with math? As an example, consider the first option listed above. Infinite mass is not mathematically inconsistent. It just means that the object is immune to external net forces — it cannot be made to accelerate. If its initial velocity is zero, then it remains zero forever no matter how hard we push on it. Mathematically, that's fine. Mathematically, an object can have a changeable orientation even if it doesn't have a changeable location.
...with physics? No real thing is known to be localized at a mathematical point (no experiment could ever verify it), but a real thing can often be localized in a region that is very small compared to other scales of interest. That's when modeling the thing as a point can be useful. In some applications, even a star can be modeled as a point! In an application where the electric dipole moment, the moment of inertia, and the mass are all important, we need to use a non-pointlike model. Sometimes people say that an electron is pointlike as far as we know (and this is true in a certain non-obvious technical sense), but the electron also doesn't have any moment of inertia (or electric dipole moment!) as far as we know.