A small beads of mass ma having charge -q is constrained to move along a froctionless wire. A positive charges Q lies at a distance L from the wire. Initiallym the bead is just above charge +Q. Show that if the bead is displaced a distance x, where `xltltL`, and released, it will exhibit simple harmonic motion. obtain an expression for the tive period of simple harnonic motion.

Answer

`2 pi sqrt((4 pi epsilon_(0)mL^(3))/(Q q))`.

Explanation

Components F cos `theta` of attractive coulomb's force restores the charge q to its mean positon O.

From Newton's second law,
`-F cos theta =ma`
or, `(Qq)/(4 pi epsilon_(0)(x^(2)+L^(2))) (x)/(sqrt(x^(2)+L^(2)))=ma`
or, `a=(Qq)/(4 pi epsilon_(0)m) (x)/((x^(2)+L^(2))^(3//2))`
For a small linear displacement , `x ltlt L`, we can ignore `x^(2)` in conparison to `L^(2)`.
`a~=-(Qq)/(4 pi epsilon_(0)mL^(3))x`
which is equation of SHM with `omega=sqrt((Qq)/(4 pi epsilon_(0)mL^(3))`
`T=(2 pi)/(omega) =2 pi sqrt((4 pi epsilon_(0)mL^(3))/(Qq))`.