Four equal positive charges, each of value Q, are arranged at the four corners of a square of diagonal 2a. A small body of mass m carrying a unit positive charge is placed at height h above the center of the square.
What should be the value of Q in order that this body may be in equilibrium?

Options

A. stable equilibrium
B. unstable equilibrium
C. neutral equilibrium
D. cannot be determined

Answer

A

Explanation

`OA=OB=OC=OD=a`
The magnitude of the electric force due to each charge is
`(1)/(4 pi epsilon_(0)) (Q)/(a^(2)+h^(2))`
The components of the force perpendicular to OP sum up to zero because of the smmetrical distribution of charge about OP. Hence, the resultalant force at P is upward along OP. The magnitude of the force is given by
`F_(up)=4xx(1)/(4 pi epsilon_(0))(Q)/(h^(2)+a^(2)) cos theta`
`(Q)/(pi epsilon_(0)(h^(2)+a^(2)))(h)/(sqrt(h^(2)+a^(2)))=(Qh)/(pi epsilon_(0)(h^(2)+a^(2))^(3//2))`
`F_("down")=weight =mg`
For equilbrium, `mg =(Qh)/(pi epsilon_(0)(h^(2)+a^(2))^(3//2))`
or `Q=pi epsilon_(0)(mg)/(h) (h^(2)+a^(2))^(3//2)`
yes this is a stabel equilibrium as it will regain its position if displaced a little.