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 ?

Explanation

As the electric field at the center of a ring is zero, the particle will not come back due to repulsion if it crosses the center (Fig. 3.64), i.e.,
.
`(1)/(2) m v^2 + (1)/(4 pi epsilon_0)(q Q)/r gt (1)/(4 pi epsilon_0)(q Q)/(R)`
But here, `Q = 2 pi R lambda` and `r = sqrt((sqrt3 R)^2 + R^2) = 2 R`
`(1)/(2) m v^2 gt (1)/(4 pi epsilon_0)(2 pi R lambda q)/(R)[1 - (1)/(2)]`
`v gt sqrt((lambda q)/(2 epsilon_0 m))`,
Therefore, `v_(min) = sqrt((lambda q)/(2 epsilon_0 m))`.