# What is dipolar charge distribution?

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?

• The 3D Dirac delta function describes the charge density of a point charge. There is nothing continuous about it. – G. Smith Apr 22 at 12:45
• 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. – G. Smith Apr 22 at 12:47
• Do you mean that dipole charge distribution is not a continuous distribution? – Alfred Apr 22 at 14:27
• Dirac $\delta$ functions are a type of distribution function (generalized function); cf. Ibragimov 2010 ch. 8. – Geremia Apr 25 at 20:51
• 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. – Emilio Pisanty May 12 at 15:46

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.