Question

Two point charges $+4q$ and $+q$ are separated by distance $r$, where should a third point charge $Q$ be placed that the whole system remains in equilibrium

• A. $\frac{2r}{3}$ from $q$
• B. $\frac{r}{3}$ from $q$
• C. $\frac{r}{2}$ from $q$
• D. $\frac{r}{3}$ from $4q$

Answer 1

The correct option is B $\frac{r}{3}$ from $q$

Suppose $Q$ is placed at distance $x$ from $4q$
$F_{Q-4q}=F_{Q-q}$
$\frac{1}{4\pi {\epsilon }_0}\frac{Q\times 4q}{x^2}=\frac{1}{4\pi {\epsilon }_0}\frac{Q\times q}{(r-x{)}^2}$
$\frac{4}{x^2}=\frac{1}{(r-x{)}^2}$
$\Rightarrow (r-x{)}^2\times 4=x^2$
Square root on both sides,
$2(r-x)=x$
$2r=3x$
$x=\frac{2}{3}r$
$\frac{2}{3}r$ from $4q$ and $(r-x)=\frac{r}{3}$ from $q$
Option (2) is correct.