Two charges -q each are fixed separated by distance 2d. A third charge q of mass m placed at the mid-point is displaced slightly by x(x<<d) perpendicular to the line joining the two fixed charged as shown in figure. Show that q will perform simple harmonic oscillation of time period.
Let us elaborate the figure fist.
Given, two charge -q at A and B
x = small distance perpendicular to O.
i.e., x<d mass of charge Q is. So, force of attraction at P towards A and B are each where AP=BP=r
Horizontal components of these forces Fn are cancelled out. Vertical components along PO add.
If the net force on q along PO is
When,
where,
i.e., force on charge q is proportional to its displacement from the centre O and it is directed towards O.
Hence, motion of charge q would be simple harmonic, where
and
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