This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
case 1)  displacement horizontally:

    If the charge q is displaced slightly to the right side along the line joining the two charges -q, then the force of attraction of the right side -q is more than the force of attraction of the -q on the left side.
  So +q moves towards right side. As it moves closer to the -q (right side), the force increases.  Then it will accelerate more and collide with it.

case 2)   displacement of +q along the perpendicular bisector of line joining two charges -q and -q.

Let the direction of x be towards the top upwards from the center.  Let the origin be at the center of two charges -q.  Let the charge q be displaced by x , where x << d.   The distance between the two charges is 2 d.  Mass of charge q  is m.
The distance between -q and +q = √(d² + x²)

Let K = 1/[ 4 πε ].

see the diagram.  Let F1 be the force due to -q on the right and F2 be the force due to -q on the left side.   They are equal in magnitude and directions are as shown.

   magnitude = F1 = F2 = K q² / (d² + x²)
Components of F1 and F2 along the perpendicular bisector are:  
          F1 Sin Ф = K q²  x / (x²+d²)³/²

 The components of these forces parallel to the line joining -q charges are = F1 Cos Ф and will cancel as they are in opposite directions.

Let F and a be the instantaneous resultant force and net acceleration of +q.  

Then the resultant force on q due to the two charges: 
      F =  2 K q² x / (d²+x²)³/²  towards the origin in the direction of - x.

     m d² x/ dt²   = - 2 K q² x / [d³ (1 + x²/d²)³/² ]
                       =  - 2 K q² x / d³   if  x << d, then we ignore x² as it is << d²

   m  d² x / d t² =  - [ 2 q² / 4 π ε d³ ]  x
                       = ⁻ω² x
  This is the equation of motion of the charge q.  SO q oscillates in a simple harmonic motion with an amplitude of  x₀ that is the displacement at t = 0.
   The angular frequency of oscillation =  ω  = q / √ (2π ε d³)

2 5 2
click on thanks (Azure blue button) above