Question

Two point charges +q and fixed at (a,0,0) and (-a,0,0). A third point charge -q is at origin. State whether ite equlilbrium is stable, unstable or neutral if it is slightly displaced: (a) along x-axis. , (b) along y-axis.

Answer 1

The equilibrium is unstable if it is slightly displaced along x-axis and it is stable if it is slightly displaced along y-axis.

Given that two point charges +q and fixed at (a,0,0) and (-a,0,0). A third point charge -q is at origin.

Consider charge -q at the origin.

• The force on -q due to +q at ( a, 0, 0) is Kq²/a² in +x direction.
• The force on -q due to +q at ( -a, 0, 0) is Kq²/a² in -x direction.
• Therefore net force on -q is 0.

(a) If it is displaced slightly along x axis towards ( +a , 0, 0) ,

• Force on -q due to +q  at ( a, 0, 0) will be more than +q at ( -a, 0, 0).
• This is because the distance will be less for ( a, 0, 0).
• Because of the increased force towards +x direction, the -q will move towards (a, 0 , 0) .
• Hence it is at unstable equilibrium along x axis.

(b)  If it is displaced slightly along +y axis

• The net force due to both the +q charges will add up to act in the -y direction.
• Therefore it is pulled back to the original equilibrium position.
• Hence it is in stable equilibrium along y axis.