Mechanical properties of electric point dipoles Point electric dipoles are usually introduced as two charges $\pm q$ at a distance $d$, then taking the simultaneous limit $d \to 0$ and $q \to \infty$ keeping the product $qd = p$ constant. In this way, we obtain a meaningful electric quantity, the dipole moment $\vec{p}$. Next, it is shown that the electric field produces a torque (with respect to the position of the dipole) $\vec{\tau} = \vec{p} \times \vec{E}$ on the dipole.
Now, I would expect that the dipole reacts mechanically to the applied torque via the Euler's equations of motion, which contain the moment of inertia of the dipole. Two separated charges of mass $m$ obviously have a well defined moment of inertia. However, in the limit $d \to 0$, the moment of inertia vanishes (after all, we are talking about a point particle). Therefore, this limit seems problematic from a mechanical point of view.
My question: is there a sensible way to introduce a point electric dipole with well defined electrical and mechanical properties? If no, can we conclude that point electric dipoles are not consistent constructs in physics? (Luckily, no one has observed one yet!)
 A: 
is there a sensible way to introduce a point electric dipole with well defined electrical and mechanical properties?

Partly. We have a few options:

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*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:

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*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?

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*...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.
A: See, when we are talking about point dipoles we have usually pictures of polar molecules in our mind where $d \approx  1 \overset{°}{\text{A}}$ and charge separation $q \approx 10^{-10}~\textrm{esu}$. So, their dipole moment is very small.
When an electric field is applied it produces a small torque. Its moment of inertia being also infinitesimally small so it results in finite angular acceleration .
