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An electric dipole is a system of two opposite point charges when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains finite.

  1. How can we have a continuous volume charge distribution from such a collection of point charges?

  2. In the article 'Electric dipole' at Knowino it is said that

    The charge distribution is written in terms of Dirac delta functions: $$\rho (\mathbf{r})=q_1 \delta (\mathbf{r}-\mathbf{r}_1)+q_2 \delta (\mathbf{r}-\mathbf{r}_2)$$

Here $\mathbf{r}_1$ and $\mathbf{r}_2$ are the position vectors of $q_1$ and $q_2$ and $\mathbf{r}=\mathbf{r}_1-\mathbf{r}_2$. Please explain why do we need Dirac delta in describing dipolar charge distribution?

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  • $\begingroup$ The 3D Dirac delta function describes the charge density of a point charge. There is nothing continuous about it. $\endgroup$ – G. Smith Apr 22 at 12:45
  • $\begingroup$ Two point charges have a dipole moment but are not technically a dipole. A true dipole is when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains finite. $\endgroup$ – G. Smith Apr 22 at 12:47
  • $\begingroup$ Do you mean that dipole charge distribution is not a continuous distribution? $\endgroup$ – Alfred Apr 22 at 14:27
  • $\begingroup$ Dirac $\delta$ functions are a type of distribution function (generalized function); cf. Ibragimov 2010 ch. 8. $\endgroup$ – Geremia Apr 25 at 20:51
  • $\begingroup$ Note that you're misinterpreting the role of $\mathbf r$ in that formula - it represents the point at which the charge density $\rho(\mathbf r)$ is being evaluated, and not any intrinsic quantity of the charge density itself. $\endgroup$ – Emilio Pisanty May 12 at 15:46
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You don't need Dirac delta functions to describe a dipolar charge distribution. There's a very wide range of charge distributions whose electric fields are exactly dipolar (this Q&A describes one such example) and an even wider class whose electric fields are dominated by a dipolar asymptotic when you're away from the support of the charge distribution (basically: every neutral charge distribution with a nonzero dipole moment).

The expression you've asked about is simply the correct encapsulation into a single analytical formula for the charge density $\rho(\mathbf r)$ that corresponds to two point charges $q_1$ and $q_2$ at positions $\mathbf r_1$ and $\mathbf r_2$. This needs to be fairly singular (in technical language, it's a distribution, not a function) because point charges are not strictly describable as (continuous) volumetric charge densities.

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