I am currently working on a homework question and I realized that I don't really understand some of the material that we covered so I hoped that you guys can help me clear up some misunderstandings.
In the diagram I drew (the best I could do with a mouse and paint), there are two positive point charges $Q$ at $y_1=\frac{ab}{2}$ and $y_2=-\frac{ab}{2}$. A microsphere is placed at distance $ab$ from the origin. The microsphere has the charge $-q$ and mass $m$. Note that $q<<Q$
I want to find the equation of motion of the Microsphere and some numerical solutions to this equation.
This is what I got so far:
The two point charges are going to create a field that attracts the microsphere $q$. That field will exert some Force $F$ on $q$.
Since $F=ma$ I can find the equation of motion if I know the force acting on q.
Assuming I know the value of the electrical field $\vec{E}$ at all points then the force acting on a charge $q$ in that field is:
$$\vec{F}=q\vec{E}$$
I also know that because of the superposition principle the electrical field $\vec{E}$ created by two charges is just the vector sum of those two fields.
$$\implies \vec{E}_{total}=\vec{E}_1+\vec{E}_2$$
In addition I know the field created by a single charge is:
$$\vec{E}=k\frac{Q}{(\vec{r})^2}$$
I feel like I have all the information I need to solve this problem but I am somehow too dumb to put all of this together. Can somebody maybe give me a few hints.
Thanks in advance