Equal charges `q` are placed at the four corners `A,B,C,D`of a square of length `a`. The magnitude of the force on the charge at `B` will be

Options

A. `(3q^(2))/(4 pi epsilon_(0) a^(2))`
B. `(4q^(2))/(4 pi epsilon_(0) a^(2))`
C. `((1 + 2 sqrt(2))/(2)) (q^(2))/(4 pi epsilon_(0)a^(2))`
D. `(sqrt(2) + (1)/(sqrt(2))) (q^(2))/(4 pi epsilon_(0)a^(2))`

Answer

3

Explanation

Force on `q at B`:
Due to `q to A : F_(1) = (1)/(4 pi in_(0)) (q^(2))/(a^(2))` , along `AB`
Due to `q at D : F_(2) = (1)/(4 pi in_(0)) (q^(2))/(a^(2))` , along `DB`
Due to `q at C : F_(3) = (1)/(4 pi in_(0)) (q^(2))/((sqrt(2) a)^(2)) , CB`
The resultant of `F_(1)` and `F_(2) , F' = sqrt(2) . (1)/(4 pi in_(0)) (q^(2))/(a^(2))` along `CB`
Resultant force on `q at B`
`F_(0) = F' + F_(3) = (1)/(4 pi in_(0)) (q^(2))/(q^(2)) [sqrt(2) + (1)/(2)]`
`=((1 + 2sqrt(2))/(2)) (q^(2))/(4 pi in_(0) a^(2))`