Question

A charge Q is placed at each of the two opposite corners of a square. A charge q is placed at each of the other two comers. If the resultant force on each charge q is zero, then :

• A. $q=\sqrt{2}Q$
• B. $q=-\sqrt{2}Q$
• C. $q=2\sqrt{2}Q$
• D. $q=-2\sqrt{2}Q$

Answer 1

The correct option is D $q=-2\sqrt{2}Q$

The net force on q at one corner is zero if $\overrightarrow{F_1}+\overrightarrow{F_2}+\overrightarrow{F_3}=0$
or $F_1\cos {45}^o\hat{i}-F_1\sin {45}^o\hat{j}-F_2\hat{j}+F_3\hat{i}=0$
so, $F_1\cos {45}^o=-F_3....\left(1\right)$ and $F_1\sin {45}^o=-F_2....\left(2\right)$
using (1) , $\frac{k{q}^2}{(\sqrt{2}a{)}^2}\times \frac{1}{\sqrt{2}}=-\frac{kqQ}{a^2}$
or $q=-2\sqrt{2}Q$