Question

Three charges $-q,+q$ and $-q$ are placed at the corners of an equilateral triangle of side a. The resultant electric force on a charge $+q$ placed at the centroid O of the triangle is:

• A. $\frac{3q^2}{4\pi {\epsilon }_0{a}^2}$
• B. $\frac{q^2}{4\pi {\epsilon }_0{a}^2}$
• C. $\frac{q^2}{2\pi {\epsilon }_0{a}^2}$
• D. $\frac{3q^2}{2\pi {\epsilon }_0{a}^2}$

Answer 1

Answer: (D)

$\frac{3q^2}{2\pi {\epsilon }_0{a}^2}$

$cos{30}^0=\frac{a}{2x}$
$x=\frac{2a}{2\sqrt{3}}$
$x=\frac{a}{\sqrt{3}}$
The net force on x axis is cancelled by the below two ($-q$) charges.
$\therefore$ Force on y axis adds up

$F_y=\frac{k\left(q\right)\left(q\right)}{(\frac{a}{\sqrt{3}}{)}^2}+\frac{k\left(q\right)\left(q\right)}{(\frac{a}{\sqrt{3}}{)}^2}\times cos{60}^0\times 2$

$F_y=\frac{3k{q}^2}{a^2}+\frac{3k{q}^2}{a^2}$

$F_y=\frac{6k{q}^2}{a^2}$

$F_y=\frac{6q^2}{4a^2\pi {\epsilon }_0}=\frac{3q^2}{2a^2\pi {\epsilon }_0}$