Question
Does the force on a charge due to another charge depend on the charges present nearby?

Answer

Coulomb's Law states that the force between two charged particle is given by,
$\text{F}=\frac{\text{q}_1\text{q}_2}{4\pi\in_0\text{r}^2}$
Where,
$q_1$ and $q_2$ are the charges on the charged particles .
r = separation between the charged particles.
$\in_0$ = parmittivity of free space.
According to the Law of Superposition, the electrostatic force between two charged particles are unaffected due to the presence of other charges.

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