Electric Field of a Pure Dipole(Math Stuff)

[PowerWill]

Working on Griffiths 3.33. I'm supposed to show that the Electric Field of a pure dipole can be written in the following coordinate free form:
<br /> \vec{E}(\vec{r}) = \frac{1}{4 \pi \epsilon_0 r^3} [3(\vec{p} \cdot \hat{r})\hat{r} - \vec{p}]<br />
Where p is the dipole. I know that the potential is equal to
<br /> V(r,\theta) = \frac{\hat{r} \cdot \vec{p}}{4 \pi \epsilon_0 r^2}<br />
and I tried to take the negative gradient of that, but got lost in the math. If you assume the dipole points along the z-axis you get the solution
<br /> \vec{E}(r,\theta) = \frac{p(2cos \theta \hat{r} + sin \theta \hat{\theta})}{4 \pi \epsilon_0 r^3}<br />
And I tried to work with that a little to no avail. Any ideas how to solve this beast?

 

[dextercioby]

Yes,there's one advice for each part:don't get lost in maths.And use spherical components for the 2 vectors involved in the scalar product of the second...

Daniel.

 

[PowerWill]

I'm still rather confused...should I calculate the dot product by components or say it equals pcos \acute{\theta} and then try to find some weird relation between theta prime and theta? Or perhaps I'm missing something? Cuz either way I keep getting lost.

 

[dextercioby]

What primes are u talking about...?There is no prime in your equations.

As for incapacity of differentiation,well,that's simply bad.

Daniel.

 

[PowerWill]

Nevermind I got it...I was trying to use spherical coordinates all the way through instead of the spherical components of the rectangular coordinates

 

[vinter]

In such questions, it is almost always good to define your own coordinate systems. Here, the vecotors p, p (cross) r and (p cross (p cross r)) can serve as the ideal coordinate axes since they are mutually perpendicular and of course, you know the component of E along p and along p cross r.