The opposite corners of square carry Q charges each and the other two opposite corners of the same square carry q charges each. If the resultant force on q is zero , how are Q and q related?

Explanation

Let each side of square be x.

Diagonal ` = sqrt(x^(2) + x^(2)) = xsqrt2`
` F_1 = F_2 = (Qq)/( 4 piin _0 x^(2))`
` F_3 = (q q )/(4 piin _0( xsqrt2)^(2)) = (q^(2))/(2xx 4 piin _0 x^(2)) `
As ` F_1 and F_2` are perpendicular to each other , their resultant force,
` " " F = sqrt(F_1 ^(2) + F_2^(2) ) = sqrt(F_1^(2) + F_1^(2))`
or ` " "F = F_1 sqrt2`
As, not force on q is zero, therefore
` F_1 sqrt2 = - F_3`
` (Qqsqrt2)/(4 pi in _0 x^(2)) = (-q^(2))/(2 xx 4 piin _0 x^(2))`
` rArr " " q=- 2 sqrt2 Q `