Two indentical point charges having magnitude q each are placed as shown if fig. If we place a negative cahrge (of magnitude -q and mass m)at the midpoint of charges and displaced along the x-axis, examine whether it will perform simpe harmonic motion. If yes then find the time period of oscillation of the particle.

Explanation

Electric field at P=x is
`E=2[(1)/(4 pi epsilon_(0))(q)/((a^(2)+x^(2)))]cos theta`
`=(q)/(2pi epsilon_(0))(x)/((a^(2)+x^(2))^(3//2))`
Force on particle is
`F=-qE=(-q^(2))/(2pi epsilon_(0))(x)/((a^(2)+x^(2))^(3//2))`
For x ltlt a, particle will execute
SHM with time period `T=2pisqrt((2 pi epsilon_(0)ma^(3))/(q^(2))`.
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