System of three charges
Two charges $q_{1}$ and $q_{2}$ are located at points with position vectors $\textbf r_{1}$ and $\textbf r_{2}$ respectively. A third charge $q_{3}$ is to be placed at a position $\textbf r_{3}$ such that the system of three charges remains in equilibrium. Is it necessary for $q_{1},q_{2},q_{3}$ to be collinear?
My attempt and assumptions: Firstly, if any of the three charges has an unbalanced force, the system is then not in equilibrium. So, let us look at the equilibrium condition for charge $q_{3}$
Case 1: $q_{1},q_{2}$ have opposite polarities.
(a) $|q_{1}|=|q_{2}|$
We have,
It seems to me the polarity of $q_{3}$ is irrelevant. It only flips the direction in the electric field lines generated by $q_{1},q_{2}$ $$\textbf E(r,\theta)=\frac{1}{4\pi\epsilon_{o}}(\frac{2p\cos(\theta)}{r^3}\textbf e_{r}+\frac{p\sin(\theta)}{r^3}\textbf e_{\theta})$$ (Here, $p$ is the dipole moment)
For $\textbf E(r,\theta)=\textbf 0$
The radial and transverse components must be independently be $0$. 649
This is impossible as there is no value of $\theta$ for which $\sin$ and $\cos$ are simultaneously $0$ since $p$ cannot be $0$.
(b) $q_{1}\neq q_{2}$
I assume this only makes the field stronger with respect to the charge having higher magnitude. (I assume it to make the field lines more sharper with respect to the charge with higher magnitude), and should overall not generate an equilibrium point.
Since in case 1, $q_{3}$ itself is not in equilibrium, we cannot even talk about the whole system.
Case 2:$q_{1},q_{2}$ have same polarities.
The electric field lines suggest that the equilibrium point for $q_{3}$ lies somewhere along the line joining and in between $q_{1},q_{2}$, exactly at the midpoint if $|q_{1}|=|q_{2}|$. Along the line but not in between, $q_{3}$ will feel attraction/repulsion depending on it's polarity.
(a)If the polarity of $q_{3}$ is the same as $q_{1},q_{2}$,then, $q_{3}$ will be at equilibrium at some point along the line and in between but simultaneously $q_{1}$ and $q_{2}$ will be repelled along the same line, in direction away from $q_{3}$. So the polarity of $q_{3}$ cannot be the same as $q_{1},q_{2}$.
(b)If the polarity of $q_{3}$ is different from $q_{1},q_{2}$. This looks like the probable case to produce an equilibrium point.
Conclusion: $q_{1},q_{2}$ must have same polarity and $q_{3}$ must have different polarity. Only then, an equilibrium point can be established for $q_{3}$ lying at some point in between the line joining $q_{1},q_{2}$ 37