Modeling a pure dipole as a function similar to a Dirac delta function

Question

I am taking an undergraduate course in E&M following Griffiths.

I was wondering if there is a good way to embed the information of a dipole into the charge distribution (and if it would be of any use) as the limit of some function similar to a Dirac delta function, or modeling it in some other manner. I know that you can write a pure dipole as a limit where the distance between two point charges tends to $0$ but the dipole moment of the system does not.

My Thoughts and Ideas so far (Limited to 1D):

Using two delta functions and a limit for the distance. This solution is awkward and feels like it involves too many working parts to be a satisfactory solution - I would prefer just a single limit as opposed to three of them.

So, I was experimenting with functions like $nxe^{-nx^2}$ which have a total charge of $0$. However, this does not work since the dipole moment still tends to $0$ when $n$ tends to $\infty$.

My Question:   Is there anywhere in physics, where a charge distribution with a dipole would be used? If so, is there an agreed upon definition for such a function? (Like the Dirac delta with the Integral of Product rule and unit area property?)

If not, then is there any other way of defining/modeling a pure dipole more concretely (directly?) rather than through the multipole expansion.

Answer 1

The (distributional) derivative of the delta function does what you want. By definition, $$ \int \mathrm dx \ \delta'(x)f(x) = -f'(0)$$ which is motivated by the standard integration by parts formula. As a result, a charge distribution $\rho(\mathbf r) = \alpha \delta (x) \delta(y) \delta'(z)$ produces a net dipole moment $$\mathbf p = \int \mathrm dx \mathrm dy \mathrm dz \ \mathbf r \ \rho(\mathbf r) = -\alpha \hat z$$ and no other multipole moments. More generally, a dipole $\mathbf p = p \hat n$ (where $\hat n$ is a unit vector) is generated by the charge distribution $\rho(\mathbf r ) = - p \big(\hat n \cdot \nabla\big)\delta^3(\mathbf r) = - \mathbf p \cdot \nabla\big[\delta(x)\delta(y)\delta(z)\big] $