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A small bead of mass m having charge -q is constrained to move along a frictionless wire. A positive charge Q lies at a distance L from the wire. Initially, the bead is just above charge +Q. Show that if the bead is displaced a distance x, where x « L and released, it will exhibit simple harmonic motion. Obtained an expression for the thime period of simple harmonic motion.

A small bead of mass m having charge -q is constrained to move along a frictionless wire. A positive charge Q lies at a distance L from the wire. Initially, the bead is just above charge +Q. Show that if the bead is displaced a distance x, where x « L and released, it will exhibit simple harmonic motion. Obtained an expression for the thime period of simple harmonic motion.

Grade:11

1 Answers

Kapil Khare
80 Points
5 years ago
Force on the bead after displacement = F = -KQq/(L + x)2
 
Since L » x
F \approx -KQq/L2
 
Let this attractive force make angle \theta with the wire.
Then only cos\theta component of the force will be towards the central position.
 
Fcos\theta = (– KQq/L2) *(x/(L+x))
again L>>x 
\implies   Resultant force = -KQqx/L3
Since F \propto -x
The motion will be Simple Harmonic Motion.

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