Potential energy of a dipole point and point charge

[hansbahia]

Homework Statement

A point dipole p is at the origin of coordinates, while a point charge q is at position r. Calculate the potential energy U of this system in two ways (a) by treating the dipole as a source and q as a test charge in the dipole field (b) by treating the dipole as a test-object in the Coulomb electric field of the point-charge, now treated as a source

Homework Equations

U=-pE
V(r)=Integral of Edr

The Attempt at a Solution

I found the potential

V(r)=p.r/(4πε0r^3)=p.r(hat)/(4πε0r^2)=pcosθ/(4πε0r^2)

Then I differentiated since E(r)=d/dr(V(r))

E(r)=-d/dr(V(r))=-d/dr(pcosθ/(4πε0r^2))=-(-3pcosθ/(4πε0r^3))=3pcosθ/(4πε0r^3)

So my U would be equal

U=-pE=-p(3pcosθ/(4πε0r^3))=-3p^2cosθ/(4πε0r^3) ?

It doesn't make sense

 

[frogjg2003]

First, $\vec{E}= -\nabla \Phi \ne -\frac{\partial \Phi}{\partial r}$.
Second, the energy of a charge q in potential V is $U=qV$.
Third, the energy of a dipole p in field E is $U=-\vec{p}\cdot\vec{E}$.
For part a, you're supposed to use a charge q in a dipole field.
For part b, you're supposed to use a dipole $\vec{p}$ in point charge field.

 

[hansbahia]

Thank you!