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## Homework Statement

Show that the electric field of a "pure" dipole can be written in the coordinate-free form

[tex]

E_{dip}(r)=\frac{1}{4\pi\epsilon_0}\frac{1}{r^3}[3(\vec p\cdot \hat r)\hat r-\vec p].[/tex]

## Homework Equations

Starting from

[tex]E_{dip}(r)=\frac{p}{4\pi\epsilon_0r^3}(2\cos \hat r+\sin\theta \hat \theta)[/tex]

## The Attempt at a Solution

The equation immediately above assumes a spherical coordinate system such that p is oriented along z. We can therefore write

[tex]\vec p=p\hat z[/tex]

[tex]\hat z = \cos\theta \hat r - \sin\theta \hat \theta \implies \vec p=p\cos\theta\hat r-p\sin\theta\hat\theta[/tex]

From equation 3.102 in the book we know that [itex]\hat r\cdot \vec p=p\cos\theta[/itex]

Try as I might I don't know how to show, geometrically or via manipulation, that [tex]p\sin\theta\hat \theta=(\vec p \cdot \hat \theta)\hat \theta[/tex]. From there it's easy to get to the desired result.