Subjective Type

Four equal points charges, each of magnitude $$+Q$$, are to be placed in equilibrium at the corners of a square. What should be the magnitude and sign of the point charge that should be placed at the centre of square to do this job?

Solution

According to the given diagram:
$$F_1\& F_2$$ are the force due to corner charge and $$F_3$$ due to central charge.

For equilibrium condition
$${ F }_{ 3 }={ F }_{ 2 }+2{ F }_{ 1 }\cos { { 45 }^{ o } } \Rightarrow \cfrac { kqQ }{ { \left( 1/\sqrt { 2 } \right) }^{ 2 } } =\cfrac { kQQ }{ { \left( \sqrt { 2 } l \right) }^{ 2 } } +2\cfrac { kQQ }{ { l }^{ 2 } } \cfrac { 1 }{ \sqrt { 2 } } \Rightarrow q=\cfrac { Q\left( 2\sqrt { 2 } +1 \right) }{ 4 } \quad $$

$$Q$$ should be negative as there is attraction between $$Q$$ and $$q$$.

so $$q=-\cfrac { Q\left( 2\sqrt { 2 } +1 \right) }{ 4 } $$

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