I am trying to work out whether there is a way to calculate the electric field of a dipole from the following formula:
$$\phi(\vec{r}) = -\vec{p} \cdot\vec{\nabla}\phi_0$$
Where $\phi_0$ is the potential of a point charge $\dfrac{1}{4 \pi \epsilon_0 r}$
I am using $E=-\vec{\nabla}\phi(\vec{r})$
Is there any formula for $\vec{\nabla}(\vec{A} \cdot \vec \nabla (f(r))$ I can use?
If not why not? Any derivation I have seen returns to
$$\phi(\vec{r}) = \dfrac{\vec{p}\cdot\vec{r}}{4 \pi \epsilon_0 r^3} = \dfrac{p \cos{\theta}}{4 \pi \epsilon_0 r^2}$$
where $\theta$ is the angle between the position $\vec{r}$ and $\vec{p}$, which points along the axis of the dipole. The centre of the dipole is at the origin and we assume $r>>p$.
The electric field is usually worked out from the above formula either by considering derivatives componentwise, or by switching to spherical polar coordinates.
