Question

Three charges are placed at the vertices of an equilateral triangle. For what value of Q both sign and magnitude does the total electric field vanish at the center of the triangle? 

Answer 1

Electric field at $q_1=q$ charge

$E_1=\frac{kq}{(\frac{a}{\sqrt3})^2}=3\frac{kq}{a^2}$

$E_2=\frac{kQ}{(\frac{a}{\sqrt3})^2}=3\frac{kQ}{a^2}$

$E_3=\frac{kQ}{(\frac{a}{\sqrt3})^2}=3\frac{kQ}{a^2}$

E₂andE₃ resultant

$E'=\sqrt{E_1^2+E_2^2+2E_1E_2cos\phi}$

$\phi=60°$

$E_1=E_2=E$

$cos60°=\frac{1}{2}$

$E'=\sqrt{2}E$

$E'=\sqrt2\frac{kQ}{a^2}$

Tringle Centre all three charge electric field zero

When

$E'=E_1$

$3\frac{kq}{a^2}=3\sqrt{2}\frac{kQ}{a^2}$

$Q=\frac{q}{\sqrt2}$