In the following problem, I have already solved for the value of the potential, and I would like to tackle the extra exercise, which asks for the electric field of a point quadrupole:
At every point in the $yz$ plane, find the electric potential of the quadrupole with three collinear charges $(+q, -2q , +q)$ that lie in the $z$ axis, with the middle charge at the origin. Consider $r \gg a$, $\vec{Q}$ is the quadrupole moment, and $\theta$ is the angle between $\vec{r}$ and $\vec{Q}$.
Hint: the quadrupole moment is $Q=\frac{1}{2} \sum_i q_i (3z_i^2-r_i^2)$.
As an extra exercise, use that result to find the electric field everywhere in the $yz$ plane.
With the help of some notes from Dr. J Tatum, I was able to understand where that value of Q comes from. So, I understand the use of Legendre polynomials and the binomial expansion to find the electric potential of a quadrupole. So the potential is given by: $$ V=\frac{1}{4\pi \varepsilon_0} \frac{qa^2}{r^3} (3\cos^2 \theta -1) = \frac{1}{4\pi \varepsilon_0} \frac{2qa^2}{r^3} (P_2(\cos \theta))$$ Where $P_n(x)$ represents the Legendre polynomials. (As a side note, I think the equations 3.12.3 and 3.12.4 in Tatum's notes are wrong).
To find the electric field I tried to use $r^3=(x^2+y^2+z^2)^{3/2}$ and then calculate $E=(-\frac{\partial V}{\partial x}, -\frac{\partial V}{\partial y}, -\frac{\partial V}{\partial z} )$, but I feel that's wrong, although I don't know why.
- Should I use the Cartesian coordinates system? Cylindrical? Spherical?
- How do I find the electric field from the electric potential?
- In case this problem's suggestion is not the most standard way to find the electric field, which method is? Is there a way to find the electric field directly without having to find the electric potential first?