A ball at O is in equilibrium as it is attached with two strings AO and DO, which are tied at A and D. `AO = DO= a(sqrt5)`. The charges at A, B, C, abd D are `+q, +Q, +2Q, and -q`, respectively. Find the correct options. The ball at O is positively charged.

Options

A. The ball O cannot remain in equilibrium.
B. If the charge at C is `+Q`, the ball will remain in equilibrium.
C. The ball O will remain in equilibrium
D. If the charge at A and D and charge at B and C are interchanged, the ball will remain in equilibrium.

Answer

C, D

Explanation

c.,d.
For ball `O` (positively charged) to be in equilibrium at the middle of `BC`

`[(q)/(4piepsilon_(0)(asqrt(5))^(2))+(q)/(4piepsilon_(0)(asqrt(5))^(2))]cos theta+(Q)/(4piepsilona^(2))`
`=(2Q)/(3piepsilon_(0)a^(2))`
`q=(5sqrt(5)Q)/(2)` (as `cos theta=(a)/(asqrt(5)`)
If `q=5sqrt(5)Q//2,` ball will be in equilibrium , even if charge are interchanged (a) is wrong and (c) and (d) are correct. If charge as `C` is `+Q`, then
`(2qcostheta)/(5)+Q=Q`
or `(2qcostheta)/(5)=0`
`Qcancel=0,theta=90^(@)`, which is not possible (b) is wrong.