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Q.
Consider the charges $q$, $q$ and $-q$ placed at the vertices of an equilateral triangle of each side $l$. The sum of forces acting on each charge is
Electric Charges and Fields
Solution:
From diagram, force on $q_1(= q)$ at $A$,
$\vec{F}_{1}=\vec{F}_{12}+\vec{F}_{13}=F\hat{r}_{1}$
where $F=\frac{q^{2}}{4\pi\varepsilon_{0}l^{2}}$ and $\hat{r}_{1}$, is the unit vector along $BC$
Force on $q_{2}\left(=q\right)$ at $B$, $\vec{F}_{2}=\vec{F}_{21}+\vec{F}_{23}=F\hat{r}_{2}$
(where $ \hat{r}_{2}$, is the unit vector along $AC$)
Force on $q_3( = -q)$ at $C$,
$\vec{F}_{3}=\vec{F}_{31}+\vec{F}_{32}=\left(\sqrt{F^{2}+F^{2}+2F\,F\,cos\,60^{\circ}}\right)\hat{n}=\sqrt{3}\,F\,\hat{n}$
where $\hat{n}=$ unit vector along the direction bisecting $\angle BCA$
$\therefore \vec{F}_{1}=\vec{F}_{2}+\vec{F}_{3}=0$
