Question

Four identical charges i.e. `q` is placed at the corners of a square of side `a`. The charge `Q` that must be placed at the centre of the square such that the whole system of charges in equilibrium is
A. `(q)/(4)`
B. `(q)/(4)[2 sqrt(2) + 1]`
C. `-(q)/(4) [ 2 sqrt(2) + 1]`
D. `-(q)/(4) [3 sqrt(3) + 1]`

Answer 1

Correct Answer - 3

`q(at B)`: Electric force
Due to `q(at A)` :
`F_(1) = k.(q^(2))/(d^(2))` , along `AB`
Due to `q (at C)`:
`F_(2) = k .(q^(2))/(d^(2))` , along `CB`
Due to `q(at D) : F_(3) = k.(q^(2))/(sqrt(2) d)^(2)` ,along `DB`
Due to `Q (at O) : F_(4) = k.(qQ)/((d//sqrt(2))^(2))` , along `OB`
`F_(x) = F_(1) + (F_(3) + F_(4)) cos 45^(@) = 0`
`k.(q^(2))/(d^(2)) + [ k.(q^(2))/((sqrt(2) d)^(2)) + k. (qQ)/((d//sqrt(2))^(2))](1)/(sqrt(2)) = 0`
`q + [(q)/(2) + 2Q](1)/(sqrt(2)) = 0`
`sqrt(2) q + (q)/(2) + 2Q = 0`
`((2 sqrt(2) + 1)q)/(2) + 2Q = 0`
`Q = -((2sqrt(2) + 1)q)/(4)`