Question
Six charges of equal magnitude, 3 positive and 3 negative are to be placed on $$PQRSTU$$ corners of a regular hexagon, such that field at the centre is double that of what it would have been if only one $$+ve$$ charge is placed at $$R.$$ Which of the following arrangement of charge is possible for $$P,Q,R,S,T$$ and $$U$$ respectively.
Six charges of equal magnitude, 3 positive and 3 negative are to be placed on $$PQRSTU$$ corners of a regular hexagon, such that field at the centre is double that of what it would have been if only one $$+ve$$ charge is placed at $$R.$$ Which of the following arrangement of charge is possible for $$P,Q,R,S,T$$ and $$U$$ respectively.
A.
$$+,+,+,-,-,-$$
B.
$$-,+,+,+,-,-$$
C.
$$-,+,+,-,+,-$$
D.
$$+,-,+,-,+,-$$
Answer :
$$-,+,+,-,+,-$$
Solution :
$$\left| {\overrightarrow E } \right| = \frac{{Kq}}{{{r^2}}}$$
Electric field due to $$P$$ on $$O$$ is cancelled by electric field due to $$S$$ on $$O.$$ Similarly Electric field due to $$Q$$ on $$O$$ is cancelled by electric field due to $$T$$ and $$O.$$ The electric field due to $$R$$ on $$O$$ is in the same direction as that of $$U$$ on $$O.$$ Therefore the net electric field is $$2\overrightarrow E .$$
$$\left| {\overrightarrow E } \right| = \frac{{Kq}}{{{r^2}}}$$
Electric field due to $$P$$ on $$O$$ is cancelled by electric field due to $$S$$ on $$O.$$ Similarly Electric field due to $$Q$$ on $$O$$ is cancelled by electric field due to $$T$$ and $$O.$$ The electric field due to $$R$$ on $$O$$ is in the same direction as that of $$U$$ on $$O.$$ Therefore the net electric field is $$2\overrightarrow E .$$
