Forces between two like charges and Newton's third law

Question

Two like charges, $q_1$ and $q_2$, are placed at a certain distance from each other. Charge $q_1$ repels charge $q_2$ with a force $F$ due to electric field of $q_1$, and according to Newton's third law, charge $q_2$ exerts an equal and opposite force $F$ on charge $q_1$.

Note, I am not saying the force on $q_1$ due to the electric field of $q_2$, but simply the force due to charge $q_1$ pushing on charge $q_2$.

Does this work this way: "There are two forces on $q_1$, one from pushing $q_2$, and another due to the electric field of $q_2$"?

Answer 1

No, it doesn't work this way.

1. N3L states that interaction forces always come in pairs. Such that if object $1$ applies force $\vec{F}_{12}$ on $2$, then $2$ applies a force $\vec{F}_{21}$ on $1$ that satisfies $\vec{F}_{21}=-\vec{F}_{12}$.
2. Coulomb's law tells you how to compute either $\vec{F}_{12}$ or $\vec{F}_{21}$ for the case of static electric charges: $$ \vec{F}_{12} = \frac{q_1q_2}{4\pi\epsilon_0}\frac{\vec{r}_2-\vec{r}_1}{||\vec{r}_2-\vec{r}_1||^3} $$ $$ \vec{F}_{21} = \frac{q_1q_2}{4\pi\epsilon_0}\frac{\vec{r}_1-\vec{r}_2}{||\vec{r}_1-\vec{r}_2||^3} $$
3. Coulomb's law is compatible with N3L, so that you can verify that indeed, $\vec{F}_{12} = -\vec{F}_{21}$.