Question

A circular ring of radius R with uniform positive charge density `lambda` per unit length is located in the y z plane with its center at the origin O. A particle of mass m and positive charge q is projected from that point `p( - sqrt(3) R, 0,0)` on the negative x - axis directly toward O, with initial speed V. Find the smallest (nonzero) value of the speed such that the particle does not return to P ?

Answer 1

As the electric field at the centre of a ring is zero the particle will not come back due to repulsion if it crosses the centre fig,
`(1)/(2)mv^(2)+(1)/(4piepsilon_(0))(qQ)/(r)gt(1)/(4pi epsilon_(0))(qQ)/(R)`
But here `Q=2pi R lambda and r=sqrt((sqrt3R)^(2)+R^(2))=2R`
So `(1)/(2)mv^(2) gt(1)/(4piepsilon_(0))(2piRlambdaq)/(R)[1-(1)/(2)] or vgt sqrt(((lambdaq)/(2epsilon_(0)m)))`
So` V_("Min")=sqrt(((lambdaq)/(2epsilon_(0)m)))`