Electric Dipole

Intuition

A positive and negative charge separated by a tiny distance create a directional field that behaves differently at close vs far range.

Formal Definition

Dipole moment $\vec{p} = q\vec{d}$ from negative to positive charge.

Mathematical Formulation

\[V = \frac{1}{4\pi\varepsilon_0}\frac{\vec{p}\cdot\hat{r}}{r^2}$$ $$\vec{E} = \frac{1}{4\pi\varepsilon_0}\frac{1}{r^3}[3(\vec{p}\cdot\hat{r})\hat{r} - \vec{p}]\]

Derivation

Superpose fields of $+q$ and $-q$ at distance $r\gg d$; keep leading $1/r^2$ and $1/r^3$ terms.

Worked Example

In a uniform field, a dipole aligns with the field ($U=-pE\cos\theta$).

Common Mistakes

Confusing $p=qd$ with $p=q/2d$.
Using point-dipole formula at $r\sim d$.

Quiz

Q1. What is the far-field behavior of a dipole?

Answer

Falls as $1/r^3$.