Finding the Electric Field given the potential in spherical

[John004]

Homework Statement

The problem statement is in the attachment

Homework Equations

E[/B] = -∇φ

∇ = (∂φ/∂r)e_r

The Attempt at a Solution

I am confused about how to do the derivative apparently because the way I do it gives

E = - (∂[p*r/4πε₀r³]/∂r)e_r = 3*(p*r)/4πε₀r⁴e_r

 

[pasmith]

$\mathbf{r}$ is not a constant.

I would suggest staying in Cartesian coordinates so that $$\frac{\partial \phi}{\partial x_i} = \frac1{4\pi\epsilon_0} \sum_j p_j \frac{\partial}{\partial x_i}\left(\frac{x_j}{r^3}\right)$$ and using the results $$\frac{\partial r}{\partial x_i} = \frac{x_i}{r}$$ and $$\frac{\partial x_j}{\partial x_i} = \begin{cases} 1, & i = j, \\ 0, & i \neq j.\end{cases}$$

 

[John004]

$\mathbf{r} = r\hat{\mathbf{r}}$ is not a constant...

well if I plugged that in for r, wouldn't I just get

E = - (∂[p*re_r/4πε0r3]/∂r)er = (p*e_r)/2πε₀r³e_r ?
I haven't done vector calculus in a long time, idk if I am forgetting something obvious or what