Question

Two equal positive charges $ {{2Fcos\theta }} $ are fixed at points (a,0) and (-a,0) on the x- axis. An opposite charge $ {{ - q}} $ at rest is released from point (0, a) on the y- axis. The charge will
(A) Move to infinity
(B) Move to origin and rest there
(C) Undergo SHM about the origin
(D) Execute oscillatory periodic motion but not SHM

Answer 1

Hint: To solve this problem, we need to find the distance between both the masses. We should simply balance the forces on masses with charge $ {{Q}} $ .
We would be using the Coulomb’s law:
 $ {{F = }}\dfrac{{{{K}}{{{q}}_{{1}}}{{{q}}_{{2}}}}}{{{{{r}}^{{2}}}}} $
Here, $ {{F}} $ is the force between the masses
 $ {{k}} $ is the Coulomb’s constant
 $ {{{q}}_{{1}}}{{,}}{{{q}}_{{2}}} $ are the charges
 $ {{r}} $ is the distance between the charges.

Complete step by step answer:
We already know that, initially the charges are equidistant with the origin in the x- axis
And, a negative charge is on y- axis
Here net force acting on $ {{ - q}} $ due to positive charges is not proportional to (x)
As $ {{F}} \propto - {{x}} $
On applying Coulomb’s law,
 $ {{F = }}\dfrac{{{{2K}}{{{Q}}^{{2}}}{{x}}}}{{\left( {\dfrac{{{{{a}}^{{2}}}}}{{{4}}}} \right){{ + }}{{{x}}^{{2}}}}} $
Since, here (F) net on $ {{ - q}} $ is $ {{2Fcos}}\theta $
Now, we know that F is not proportional to x .
So, motion is not SHM.
But the motion is oscillatory as at a point F acts downwards and at other F acts upwards.
So, we need to select the correct option.
The correct option is D.

Additional Information
Coulomb's law, or Coulomb's inverse-square law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force. The law was first discovered in 1785 by French physicist Charles-Augustin de Coulomb, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism

Note:
Two equal positive charges are placed at two points on the x- axis, a charge of opposite charge that is negative charge is released at a point on the y- axis, as we know that the two unlike charges attract each other the force of attraction acts between the positive charges and the negative charge. When the charge of $ {{ - q}} $ is released, both the positive charges present along the x-axis attract the negative charge in such a way that the horizontal component of force cancels each other and the net force will be in downward direction towards the origin. When the particles cross the origin the net force acts on the particle but in vertical direction. Thus, the particle executes periodic oscillatory motion.