Find the electrostatic potential arising from an electric dipole of magnitude $d$ situated a distance $L$ from the center of a grounded conducting sphere of radius $a$. Assume the dipole axis cross the sphere's center.
Honestly, this is not a hard question, it is more about be patient, solve some cosine laws and know the right approximations that is needed to use.
But the problem is not how to solve it, i have actually tried by another way than calculate the potential of the dipoles separated and sum it (including the potential of the net charge that will arise).
I have considered to use the formula $$\vec p = \int \rho \vec r d^3x$$ (in this case, basically $\sum q z$).
Assuming, first, that the distance between the dipole's charges is $2l$.
So, considering the sphere's center as the origin, the dipole we will get is (without any approximation) $$\vec p = (2ql - \frac{qR^3}{(L+l)^2} + \frac{qR^3}{(L-l)^2} )\space \hat z$$
the net charge will be
$$\delta q = \frac{2qR}{l^2-L^2}$$
Here comes the problem:
Isn't the potential of a system, until the dipole order, $$\phi = \frac{k \space q_{net} }{r} + \frac{k \space \vec p \vec r}{r^3}$$
$$?$$
Using this as the potential, we get a answer that involve just $cos(\theta)$. But the right answer is different, and i understand how the answer the book got was arrived. My question is why i can't use the one i posted here?
EDIT: BOOK ANSWER